| import OAI.Combinatorics.CompleteCrossing.Angles |
| import OAI.Combinatorics.CompleteCrossing.Tracks |
| import OAI.Combinatorics.Crossing.AffinePathHalves |
| import OAI.Combinatorics.Crossing.SubdivisionSegments |
| import Mathlib.Algebra.BigOperators.Fin |
|
|
| namespace OAI |
|
|
| /-! Matching refined crossing pieces and arc directions with truncated angle coordinates. -/ |
|
|
| section |
|
|
| noncomputable section |
| open Set Zarankiewicz |
| namespace Paper170Geometry |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| theorem normalized_piece_vertex_not_crossing (N : Paper170.NormalizedDrawing graph) |
| (e : Edge) (i : ℕ) (hi : i ≤ (N.polygonal e).pieces) : |
| (N.polygonal e).vertices i ∉ Paper170.crossingPoints N.toContinuousDrawing := by |
| have he := (Paper170.toZarankiewiczParametrization (N.polygonal e)).vertices_at |
| (⟨i, by omega⟩ : Fin ((N.polygonal e).pieces+1)) |
| change (N.polygonal e).vertices i = _ at he |
| rw [he] |
| exact N.crossings_not_knots e _ |
|
|
| theorem normalized_crossing_unique_piece (N : Paper170.NormalizedDrawing graph) |
| (e : Edge) (x : Plane) (hx : x ∈ Paper170.crossingPoints N.toContinuousDrawing) |
| (hxe : x ∈ Paper170.edgeInterior N.toContinuousDrawing e) : |
| ∃!i : Fin (N.polygonal e).pieces, |
| x ∈ openSegmentTrace ((N.polygonal e).vertices i.val) ((N.polygonal e).vertices (i.val+1)) := by |
| obtain ⟨t, ht, rfl⟩ := hxe |
| have hn (i : Fin ((N.polygonal e).pieces+1)) : t ≠ (N.polygonal e).knot i := by |
| intro he |
| exact N.crossings_not_knots e i (he ▸ hx) |
| obtain ⟨i, hi⟩ := parametrization_knot_interval (N.polygonal e) t ht hn |
| have hmem : N.edge e t ∈ openSegmentTrace ((N.polygonal e).vertices i.val) |
| ((N.polygonal e).vertices (i.val+1)) := by |
| rw [← parametrization_open_piece] |
| exact ⟨t,hi,rfl⟩ |
| refine ⟨i,hmem,?_⟩ |
| intro j hj |
| rw [← parametrization_open_piece] at hj |
| obtain ⟨s,hs,he⟩ := hj |
| have hst : s=t := N.edge_injective e he |
| subst s |
| apply le_antisymm |
| · by_contra h |
| have hij : i.succ ≤ j.castSucc := by change i.val+1 ≤ j.val; exact lt_of_not_ge h |
| exact hi.2.not_ge (((N.polygonal e).increasing.monotone hij).trans hs.1.le) |
| · by_contra h |
| have hji : j.succ ≤ i.castSucc := by change j.val+1 ≤ i.val; exact lt_of_not_ge h |
| exact hs.2.not_ge (((N.polygonal e).increasing.monotone hji).trans hi.1.le) |
| |
| theorem openSegmentTrace_of_mem_segment_avoiding (a b x : Plane) (S : Set Plane) |
| (hx : x∈S) (ha : a∉S) (hb : b∉S) (hseg : x∈segment ℝ a b) : |
| x∈openSegmentTrace a b := by |
| rw [← Path.range_segment] at hseg |
| obtain ⟨t,ht⟩ := hseg |
| have he : segmentPoint a b t.val=x := by |
| simpa only [Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using ht |
| exact ⟨t.val,segment_parameter_strict_of_avoids_endpoints a b x S hx ha hb t.val t.property he,he⟩ |
|
|
| theorem minimum_crossing_piece_unique [Fintype Vertex] (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (e : Edge) (x : Plane) |
| (hx : x ∈ Paper170.crossingPoints N.toContinuousDrawing) |
| (i j : Fin (V.chain e).pieces) |
| (hi : x∈segment ℝ ((V.chain e).vertices i.val) ((V.chain e).vertices (i.val+1))) |
| (hj : x∈segment ℝ ((V.chain e).vertices j.val) ((V.chain e).vertices (j.val+1))) : i=j := by |
| suffices hh : ∀ i j : Fin (V.chain e).pieces, i<j → |
| x∈segment ℝ ((V.chain e).vertices i.val) ((V.chain e).vertices (i.val+1)) → |
| x∈segment ℝ ((V.chain e).vertices j.val) ((V.chain e).vertices (j.val+1)) → False by |
| rcases lt_trichotomy i j with h | h | h |
| · exact (hh i j h hi hj).elim |
| · exact h |
| · exact (hh j i h hj hi).elim |
| intro i j hij hi hj |
| by_cases he : j.val=i.val+1 |
| · have hj' : x∈segment ℝ ((V.chain e).vertices (i.val+1)) ((V.chain e).vertices (i.val+2)) := by |
| simpa only [he, Nat.add_assoc] using hj |
| have hxv := (V.chain e).minimum_adjacent_intersection (V.minimal e) i.val (by omega) hi hj' |
| exact minimum_vertices_not_crossing N.toAdmissibleDrawing (normalized_locallyStraight N) |
| e (V.chain e) (V.minimal e) (i.val+1) (by omega) (hxv ▸ hx) |
| · exact Set.disjoint_left.mp ((V.chain e).minimum_nonadjacent_disjoint (V.minimal e) |
| i.val j.val (by change i.val<j.val at hij; omega) j.isLt) hi hj |
| |
| theorem truncated_crossing_unique_piece [Fintype Vertex] (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (e : Edge) (x : Plane) |
| (hx : x ∈ Paper170.crossingPoints N.toContinuousDrawing) |
| (hxe : x ∈ Paper170.edgeInterior N.toContinuousDrawing e) : |
| ∃!i : Fin (V.pieces e), x∈openSegmentTrace (V.vertices e i.val) (V.vertices e (i.val+1)) := by |
| have hxr := crossing_mem_truncated_edge N.toAdmissibleDrawing (normalized_noAdjacent N) |
| V.chain V.minimal V.radius V.radius_pos V.weak_short V.piece_clearance e x hx hxe |
| obtain ⟨i,hi,hxi⟩ := polygonalArcPath_range_piece (V.vertices e) (V.pieces e) |
| (V.pieces_pos e) x hxr |
| have hm : x∈openSegmentTrace (V.vertices e i) (V.vertices e (i+1)) := |
| openSegmentTrace_of_mem_segment_avoiding _ _ x (Paper170.crossingPoints N.toContinuousDrawing) hx |
| (V.vertices_not_crossing (normalized_noAdjacent N) (normalized_locallyStraight N) e i (by omega)) |
| (V.vertices_not_crossing (normalized_noAdjacent N) (normalized_locallyStraight N) e (i+1) (by omega)) hxi |
| refine ⟨⟨i,hi⟩,hm,?_⟩ |
| intro j hj |
| apply minimum_crossing_piece_unique N V e x hx |
| · exact (V.chain e).truncated_piece_subset V.radius V.radius_pos (V.weak_short e).1 |
| (V.weak_short e).2 j.val j.isLt (openSegmentTrace_subset_segment _ _ hj) |
| · exact (V.chain e).truncated_piece_subset V.radius V.radius_pos (V.weak_short e).1 |
| (V.weak_short e).2 i hi hxi |
|
|
| /-- Every actual polygonal-chain piece is traversed along an increasing affine parameter interval. -/ |
| theorem polygonalArcPath_piece_interval (v : ℕ → Plane) : ∀ n i, i<n → |
| ∃a b : unitInterval, a<b ∧ ∀t : unitInterval, |
| polygonalArcPath v n (intervalSegment a b t)=segmentPoint (v i) (v (i+1)) t.val |
| | 0, i, hi => by omega |
| | 1, i, hi => by |
| have he : i=0 := by omega |
| subst i |
| refine ⟨0,1,by norm_num,?_⟩ |
| intro t |
| simp only [polygonalArcPath,intervalSegment_zero_one,Path.segment_apply, |
| AffineMap.lineMap_apply_module,segmentPoint] |
| | n+2, i, hi => by |
| by_cases hprev : i<n+1 |
| · obtain ⟨a,b,hab,hseg⟩ := polygonalArcPath_piece_interval v (n+1) i hprev |
| refine ⟨intervalLeft a,intervalLeft b,intervalLeft_strictMono hab,?_⟩ |
| intro t |
| rw [polygonalArcPath,intervalSegment_left,path_trans_intervalLeft] |
| exact hseg t |
| · have he : i=n+1 := by omega |
| subst i |
| refine ⟨intervalRight 0,intervalRight 1,intervalRight_strictMono (by norm_num),?_⟩ |
| intro t |
| rw [polygonalArcPath,intervalSegment_right,path_trans_intervalRight,intervalSegment_zero_one] |
| simp only [Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Zarankiewicz |
| namespace Paper170Geometry |
| variable {Vertex Edge : Type} [Fintype Vertex] {graph : Paper170.Graph Vertex Edge} |
|
|
| /-- At an actual crossing, the truncated chain and original subdivision point in the same direction. -/ |
| theorem truncated_direction_pos_mul_original (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (e : Edge) |
| (i : Fin (V.pieces e)) (k : Fin (N.polygonal e).pieces) (x : Plane) |
| (hx : x∈Paper170.crossingPoints N.toContinuousDrawing) |
| (hxi : x∈openSegmentTrace (V.vertices e i.val) (V.vertices e (i.val+1))) |
| (hxk : x∈openSegmentTrace ((N.polygonal e).vertices k.val) ((N.polygonal e).vertices (k.val+1))) : |
| ∃r : ℝ, 0<r ∧ V.vertices e (i.val+1)-V.vertices e i.val = |
| r • ((N.polygonal e).vertices (k.val+1)-(N.polygonal e).vertices k.val) := by |
| have hfull := (V.chain e).truncated_piece_subset V.radius V.radius_pos |
| (V.weak_short e).1 (V.weak_short e).2 i.val i.isLt (openSegmentTrace_subset_segment _ _ hxi) |
| have hxfull := minimum_piece_crossing_open N.toAdmissibleDrawing (normalized_locallyStraight N) |
| e (V.chain e) (V.minimal e) i.val i.isLt x hx hfull |
| obtain ⟨a,b,hab,hseg⟩ := polygonalArcPath_piece_interval (V.chain e).vertices |
| (V.chain e).pieces i.val i.isLt |
| have hend : N.vertex (graph.src e) ≠ N.vertex (graph.dst e) := |
| fun he => graph.loopless e (N.vertex_injective he) |
| have hγseg : ∀t : unitInterval, |
| N.edge e (intervalSegment ((N.polygonal e).knot k.castSucc) ((N.polygonal e).knot k.succ) t) = |
| Paper170.segmentPoint ((N.polygonal e).vertices k.val) ((N.polygonal e).vertices (k.val+1)) t.val := |
| (Paper170.toZarankiewiczParametrization (N.polygonal e)).segment_formula k |
| have hηseg : ∀t : unitInterval, (V.chain e).path (intervalSegment a b t) = |
| Paper170.segmentPoint ((V.chain e).vertices i.val) ((V.chain e).vertices (i.val+1)) t.val := hseg |
| obtain ⟨r,hr,he⟩ := Paper170.path_segments_direction_of_mem (N.edge e) (V.chain e).path |
| (N.edge_injective e) ((V.chain e).minimum_path_injective hend (V.minimal e)) |
| (drawing_chain_range_eq N.toContinuousDrawing e (V.chain e)).symm |
| ((N.polygonal e).knot k.castSucc) ((N.polygonal e).knot k.succ) a b |
| ((N.polygonal e).increasing Fin.castSucc_lt_succ) hab |
| ((N.polygonal e).vertices k.val) ((N.polygonal e).vertices (k.val+1)) |
| ((V.chain e).vertices i.val) ((V.chain e).vertices (i.val+1)) hγseg hηseg x hxk hxfull |
| refine ⟨(V.chain e).truncatedScale V.radius i.val * r, |
| mul_pos ((V.chain e).truncatedScale_pos V.radius V.radius_pos (V.short e).1 (V.short e).2 i.val) hr, ?_⟩ |
| change trackDirection ((V.chain e).truncatedVertices V.radius) i.val = _ |
| rw [(V.chain e).truncated_direction V.radius i.val i.isLt] |
| change (V.chain e).truncatedScale V.radius i.val • |
| ((V.chain e).vertices (i.val+1)-(V.chain e).vertices i.val) = _ |
| rw [he,smul_smul] |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Zarankiewicz |
| namespace Paper170Geometry |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| theorem minimum_first_direction (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (e : Edge) : |
| ∃r : ℝ, 0 < r ∧ (V.chain e).vertices 1 - (V.chain e).vertices 0 = |
| r • ((N.polygonal e).vertices 1 - (N.polygonal e).vertices 0) := by |
| let k : Fin (N.polygonal e).pieces := ⟨0, (N.polygonal e).positive⟩ |
| obtain ⟨a,b,hab,hseg⟩ := polygonalArcPath_piece_interval (V.chain e).vertices |
| (V.chain e).pieces 0 (V.pieces_pos e) |
| have hend : N.vertex (graph.src e) ≠ N.vertex (graph.dst e) := |
| fun he => graph.loopless e (N.vertex_injective he) |
| apply Paper170.path_segments_direction (N.edge e) (V.chain e).path |
| (N.edge_injective e) ((V.chain e).minimum_path_injective hend (V.minimal e)) |
| (drawing_chain_range_eq N.toContinuousDrawing e (V.chain e)).symm |
| ((N.polygonal e).knot k.castSucc) ((N.polygonal e).knot k.succ) a b |
| ((N.polygonal e).increasing Fin.castSucc_lt_succ) hab |
| ((N.polygonal e).vertices 0) ((N.polygonal e).vertices 1) |
| ((V.chain e).vertices 0) ((V.chain e).vertices 1) |
| ((Paper170.toZarankiewiczParametrization (N.polygonal e)).segment_formula k) |
| hseg 0 0 (by norm_num) (by norm_num) |
| simpa [Paper170.segmentPoint] using |
| (N.polygonal e).vertices_first.trans (V.chain e).first.symm |
|
|
| theorem minimum_last_direction (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (e : Edge) : |
| ∃r : ℝ, 0 < r ∧ (V.chain e).vertices ((V.chain e).pieces-1) - |
| (V.chain e).vertices (V.chain e).pieces = |
| r • ((N.polygonal e).vertices ((N.polygonal e).pieces-1) - |
| (N.polygonal e).vertices (N.polygonal e).pieces) := by |
| have hn : 0 < (N.polygonal e).pieces := (N.polygonal e).positive |
| have hv : 0 < (V.chain e).pieces := V.pieces_pos e |
| let k : Fin (N.polygonal e).pieces := ⟨(N.polygonal e).pieces-1, by omega⟩ |
| obtain ⟨a,b,hab,hseg⟩ := polygonalArcPath_piece_interval (V.chain e).vertices |
| (V.chain e).pieces ((V.chain e).pieces-1) (by omega) |
| have hend : N.vertex (graph.src e) ≠ N.vertex (graph.dst e) := |
| fun he => graph.loopless e (N.vertex_injective he) |
| have hnk : k.val+1 = (N.polygonal e).pieces := by dsimp [k]; omega |
| have hvk : (V.chain e).pieces-1+1 = (V.chain e).pieces := by omega |
| have hγseg := (Paper170.toZarankiewiczParametrization (N.polygonal e)).segment_formula k |
| change ∀t, N.edge e (intervalSegment ((N.polygonal e).knot k.castSucc) |
| ((N.polygonal e).knot k.succ) t) = |
| Paper170.segmentPoint ((N.polygonal e).vertices k.val) |
| ((N.polygonal e).vertices (k.val+1)) t.val at hγseg |
| rw [hnk] at hγseg |
| rw [hvk] at hseg |
| exact Paper170.path_final_segments_direction (N.edge e) (V.chain e).path |
| (N.edge_injective e) ((V.chain e).minimum_path_injective hend (V.minimal e)) |
| (drawing_chain_range_eq N.toContinuousDrawing e (V.chain e)).symm |
| ((N.polygonal e).knot k.castSucc) ((N.polygonal e).knot k.succ) a b |
| ((N.polygonal e).increasing Fin.castSucc_lt_succ) hab |
| ((N.polygonal e).vertices k.val) ((N.polygonal e).vertices (N.polygonal e).pieces) |
| ((V.chain e).vertices ((V.chain e).pieces-1)) ((V.chain e).vertices (V.chain e).pieces) |
| hγseg hseg ((N.polygonal e).vertices_last.trans (V.chain e).last.symm) |
|
|
| /-- The truncation port lies on the positive ray of the original outward spoke. -/ |
| theorem spokePort_pos_mul_outward (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (j : Vertex) (e : IncidentEdges graph j) : |
| ∃r : ℝ, 0 < r ∧ V.spokePort j e - N.vertex j = r • Paper170.outwardSpoke N j e.val := by |
| by_cases hsrc : j = graph.src e.val |
| · obtain ⟨r,hr,he⟩ := minimum_first_direction N V e.val |
| have hp := radialPort_parameter_between ((V.chain e.val).vertices 0) |
| ((V.chain e.val).vertices 1) V.radius V.radius_pos (V.weak_short e.val).1 |
| refine ⟨(V.radius / Real.sqrt (squaredLength |
| ((V.chain e.val).vertices 1 - (V.chain e.val).vertices 0))) * r, |
| mul_pos hp.1 hr, ?_⟩ |
| change drawingSpokePort N.toContinuousDrawing V.chain V.radius j e - N.vertex j = _ |
| rw [drawingSpokePort, ite_eq_left hsrc, PolygonalChain.startPort] |
| have hcenter : N.vertex j = (V.chain e.val).vertices 0 := by |
| exact (congrArg N.vertex hsrc).trans (V.chain e.val).first.symm |
| rw [hcenter, radialPort_sub, he, smul_smul, Paper170.outwardSpoke, ite_eq_left hsrc.symm] |
| · have hdst : j = graph.dst e.val := e.property.resolve_left hsrc |
| obtain ⟨r,hr,he⟩ := minimum_last_direction N V e.val |
| have hshort : V.radius^2 < squaredLength |
| ((V.chain e.val).vertices ((V.chain e.val).pieces-1) - |
| (V.chain e.val).vertices (V.chain e.val).pieces) := by |
| rw [squaredLength_sub_comm] |
| exact (V.weak_short e.val).2 |
| have hp := radialPort_parameter_between ((V.chain e.val).vertices (V.chain e.val).pieces) |
| ((V.chain e.val).vertices ((V.chain e.val).pieces-1)) V.radius V.radius_pos hshort |
| refine ⟨(V.radius / Real.sqrt (squaredLength |
| ((V.chain e.val).vertices ((V.chain e.val).pieces-1) - |
| (V.chain e.val).vertices (V.chain e.val).pieces))) * r, mul_pos hp.1 hr, ?_⟩ |
| change drawingSpokePort N.toContinuousDrawing V.chain V.radius j e - N.vertex j = _ |
| rw [drawingSpokePort, ite_eq_right hsrc, PolygonalChain.endPort] |
| have hcenter : N.vertex j = (V.chain e.val).vertices (V.chain e.val).pieces := by |
| exact (congrArg N.vertex hdst).trans (V.chain e.val).last.symm |
| rw [hcenter, radialPort_sub, he, smul_smul, Paper170.outwardSpoke, |
| ite_eq_right (Ne.symm hsrc)] |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Circle coordinates chosen with the very same cuts and local orders as the |
| original normalized drawing. -/ |
| noncomputable section |
| open Set Zarankiewicz |
| namespace Paper170Geometry |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| theorem exists_truncatedSpokeCoordinates_compatible (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (O : Paper170.SpokeOrders N) : |
| ∃C : TruncatedSpokeCoordinates V, ∀j (e f : IncidentEdges graph j), |
| orderSign (C.node j e) (C.node j f) = Paper170.spokeComparison O j e.val f.val := by |
| classical |
| let p (j : Vertex) (e : IncidentEdges graph j) : Plane := |
| V.radius⁻¹ • (V.spokePort j e-N.vertex j) |
| have hinc (j : Vertex) (e : IncidentEdges graph j) : |
| graph.src e.val=j ∨ graph.dst e.val=j := e.property.imp Eq.symm Eq.symm |
| have hp (j : Vertex) (e : IncidentEdges graph j) : Paper170.squaredLength (p j e)=1 := by |
| change squaredLength (V.radius⁻¹ • (V.spokePort j e-N.vertex j))=1 |
| rw [squaredLength_smul,V.spokePort_circle,inv_pow, |
| inv_mul_cancel₀ (pow_ne_zero 2 V.radius_pos.ne')] |
| have hangle (j : Vertex) (e : IncidentEdges graph j) : |
| Paper170.spokeAngle (O.cut j) (p j e)= |
| Paper170.spokeAngle (O.cut j) (Paper170.outwardSpoke N j e.val) := by |
| dsimp only [p] |
| rw [Paper170.spokeAngle_smul _ _ (inv_pos.mpr V.radius_pos)] |
| obtain ⟨r,hr,he⟩ := spokePort_pos_mul_outward N V j e |
| rw [he,Paper170.spokeAngle_smul _ _ hr] |
| have hex (j : Vertex) (e : IncidentEdges graph j) : |
| ∃t : ℝ, circlePoint t=Paper170.angleRotation (O.cut j) (p j e) := by |
| apply Paper170.angleRotation_parameter _ _ (hp j e) |
| rw [hangle] |
| exact O.cut_avoids j ⟨e.val,hinc j e⟩ |
| choose node hnode using hex |
| have hpoint (j : Vertex) (e : IncidentEdges graph j) : |
| similarityPoint (N.vertex j) (Paper170.angleCoordinateFrame (O.cut j) V.radius) |
| (circlePoint (node j e))=V.spokePort j e := by |
| rw [hnode] |
| exact Paper170.angleCoordinateFrame_point _ _ V.radius_pos _ _ |
| let C : TruncatedSpokeCoordinates V := { |
| frame := fun j => Paper170.angleCoordinateFrame (O.cut j) V.radius |
| frame_ne_zero := fun j => Paper170.angleCoordinateFrame_ne_zero _ _ V.radius_pos |
| frame_length := fun j => Paper170.angleCoordinateFrame_length (O.cut j) V.radius |
| node := node |
| node_injective := by |
| intro j e f he |
| apply V.spokePort_injective (normalized_noAdjacent N) j |
| rw [← hpoint j e,← hpoint j f,he] |
| point_eq := hpoint } |
| refine ⟨C,?_⟩ |
| have harg (j : Vertex) (e : IncidentEdges graph j) : |
| Paper170.spokeAngle (O.cut j) (Paper170.outwardSpoke N j e.val) = |
| Complex.arg (Paper170.complexPoint (circlePoint (node j e))) := by |
| rw [hnode,Paper170.angleRotation_complex] |
| exact (hangle j e).symm |
| have hlt (j : Vertex) (e f : IncidentEdges graph j) : |
| @LT.lt _ (O.order j).toLT ⟨e.val,hinc j e⟩ ⟨f.val,hinc j f⟩ ↔ node j e < node j f := by |
| rw [O.compatible,harg j e,harg j f,Paper170.circlePoint_arg_lt] |
| intro j e f |
| change orderSign (node j e) (node j f)=_ |
| unfold Paper170.spokeComparison |
| rw [dite_eq_left (hinc j e),dite_eq_left (hinc j f)] |
| simp only [orderSign,hlt] |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Zarankiewicz |
| open scoped BigOperators |
| namespace Paper170Geometry |
| variable {Vertex Edge : Type} [Fintype Vertex] {graph : Paper170.Graph Vertex Edge} |
|
|
| /-- The only remaining geometric input is equality of the local oriented crossing signs. -/ |
| theorem signedIntersection_eq_of_matching_signs (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (e f : Edge) (hef : e≠f) |
| (hmatch : ∀ (i : Fin (V.pieces e)) (j : Fin (V.pieces f)) |
| (k : Fin (N.polygonal e).pieces) (l : Fin (N.polygonal f).pieces) (x : Plane), |
| x∈openSegmentTrace (V.vertices e i.val) (V.vertices e (i.val+1)) → |
| x∈openSegmentTrace (V.vertices f j.val) (V.vertices f (j.val+1)) → |
| x∈openSegmentTrace ((N.polygonal e).vertices k.val) ((N.polygonal e).vertices (k.val+1)) → |
| x∈openSegmentTrace ((N.polygonal f).vertices l.val) ((N.polygonal f).vertices (l.val+1)) → |
| segmentCrossingSign (V.vertices e i.val) (V.vertices e (i.val+1)) |
| (V.vertices f j.val) (V.vertices f (j.val+1)) = |
| segmentCrossingSign ((N.polygonal e).vertices k.val) ((N.polygonal e).vertices (k.val+1)) |
| ((N.polygonal f).vertices l.val) ((N.polygonal f).vertices (l.val+1))) : |
| V.signedIntersection e f=Paper170.interiorIntersection N e f := by |
| classical |
| let A := Fin (V.pieces e) × Fin (V.pieces f) |
| let B := Fin (N.polygonal e).pieces × Fin (N.polygonal f).pieces |
| let F : A → ℝ := fun a => segmentCrossingSign |
| (V.vertices e a.1.val) (V.vertices e (a.1.val+1)) |
| (V.vertices f a.2.val) (V.vertices f (a.2.val+1)) |
| let G : B → ℝ := fun b => segmentCrossingSign |
| ((N.polygonal e).vertices b.1.val) ((N.polygonal e).vertices (b.1.val+1)) |
| ((N.polygonal f).vertices b.2.val) ((N.polygonal f).vertices (b.2.val+1)) |
| let R : A → B → Prop := fun a b => ∃x : Plane, |
| x∈openSegmentTrace (V.vertices e a.1.val) (V.vertices e (a.1.val+1)) ∧ |
| x∈openSegmentTrace (V.vertices f a.2.val) (V.vertices f (a.2.val+1)) ∧ |
| x∈openSegmentTrace ((N.polygonal e).vertices b.1.val) ((N.polygonal e).vertices (b.1.val+1)) ∧ |
| x∈openSegmentTrace ((N.polygonal f).vertices b.2.val) ((N.polygonal f).vertices (b.2.val+1)) |
| have hsame (a : A) (b : B) (h : R a b) : F a=G b := by |
| obtain ⟨x,hx₁,hx₂,hx₃,hx₄⟩ := h |
| exact hmatch a.1 a.2 b.1 b.2 x hx₁ hx₂ hx₃ hx₄ |
| have hF : ∀a, F a≠0 → ∃!b, G b≠0 ∧ R a b := by |
| intro a ha |
| have hproper := (segmentCrossingSign_ne_zero_iff _ _ _ _).mp ha |
| obtain ⟨s,hs,t,ht,he⟩ := hproper.2 |
| let x := segmentPoint (V.vertices e a.1.val) (V.vertices e (a.1.val+1)) s |
| have hx₁ : x∈openSegmentTrace (V.vertices e a.1.val) (V.vertices e (a.1.val+1)) := ⟨s,hs,rfl⟩ |
| have hx₂ : x∈openSegmentTrace (V.vertices f a.2.val) (V.vertices f (a.2.val+1)) := ⟨t,ht,he.symm⟩ |
| have hxe := V.range_subset_edgeInterior e (V.piece_subset_range e a.1.val a.1.isLt |
| (openSegmentTrace_subset_segment _ _ hx₁)) |
| have hxf := V.range_subset_edgeInterior f (V.piece_subset_range f a.2.val a.2.isLt |
| (openSegmentTrace_subset_segment _ _ hx₂)) |
| have hx : x∈Paper170.crossingPoints N.toContinuousDrawing := ⟨e,f,hef,hxe,hxf⟩ |
| obtain ⟨i,hi,hu⟩ := normalized_crossing_unique_piece N e x hx hxe |
| obtain ⟨j,hj,hv⟩ := normalized_crossing_unique_piece N f x hx hxf |
| have hR : R a (i,j) := ⟨x,hx₁,hx₂,hi,hj⟩ |
| refine ⟨(i,j),⟨?_,hR⟩,?_⟩ |
| · rwa [← hsame a (i,j) hR] |
| · rintro b ⟨_,y,hy₁,hy₂,hy₃,hy₄⟩ |
| have hyx : y=x := openSegments_inter_subsingleton _ _ _ _ hproper.1 ⟨hy₁,hy₂⟩ ⟨hx₁,hx₂⟩ |
| subst y |
| exact Prod.ext (hu b.1 hy₃) (hv b.2 hy₄) |
| have hG : ∀b, G b≠0 → ∃!a, F a≠0 ∧ R a b := by |
| intro b hb |
| have hproper := (segmentCrossingSign_ne_zero_iff _ _ _ _).mp hb |
| obtain ⟨s,hs,t,ht,he⟩ := hproper.2 |
| let x := segmentPoint ((N.polygonal e).vertices b.1.val) ((N.polygonal e).vertices (b.1.val+1)) s |
| have hx₃ : x∈openSegmentTrace ((N.polygonal e).vertices b.1.val) |
| ((N.polygonal e).vertices (b.1.val+1)) := ⟨s,hs,rfl⟩ |
| have hx₄ : x∈openSegmentTrace ((N.polygonal f).vertices b.2.val) |
| ((N.polygonal f).vertices (b.2.val+1)) := ⟨t,ht,he.symm⟩ |
| have hxe : x∈Paper170.edgeInterior N.toContinuousDrawing e := |
| (N.polygonal e).segment_mem_interior N.toContinuousDrawing e b.1 s hs |
| have hxf : x∈Paper170.edgeInterior N.toContinuousDrawing f := by |
| rw [show x=segmentPoint ((N.polygonal f).vertices b.2.val) |
| ((N.polygonal f).vertices (b.2.val+1)) t from he] |
| exact (N.polygonal f).segment_mem_interior N.toContinuousDrawing f b.2 t ht |
| have hx : x∈Paper170.crossingPoints N.toContinuousDrawing := ⟨e,f,hef,hxe,hxf⟩ |
| obtain ⟨i,hi,hu⟩ := truncated_crossing_unique_piece N V e x hx hxe |
| obtain ⟨j,hj,hv⟩ := truncated_crossing_unique_piece N V f x hx hxf |
| have hR : R (i,j) b := ⟨x,hi,hj,hx₃,hx₄⟩ |
| refine ⟨(i,j),⟨?_,hR⟩,?_⟩ |
| · rwa [hsame (i,j) b hR] |
| · rintro a ⟨_,y,hy₁,hy₂,hy₃,hy₄⟩ |
| have hyx : y=x := openSegments_inter_subsingleton _ _ _ _ hproper.1 ⟨hy₃,hy₄⟩ ⟨hx₃,hx₄⟩ |
| subst y |
| exact Prod.ext (hu a.1 hy₁) (hv a.2 hy₂) |
| have hsum : ∑a, F a = ∑b, G b := |
| sum_eq_sum_of_unique_match F G R hF hG (fun a b _ _ h => hsame a b h) |
| rw [VertexTruncation.signedIntersection,ite_eq_right hef,Paper170.interiorIntersection,ite_eq_right hef] |
| dsimp only [A,B,F,G] at hsum |
| simp only [Fintype.sum_prod_type] at hsum |
| simp only [Paper170.segmentIntersection_eq_segmentCrossingSign] |
| change polygonIntersection (V.vertices e) (V.vertices f) (V.pieces e) (V.pieces f) = |
| polygonIntersection (N.polygonal e).vertices (N.polygonal f).vertices |
| (N.polygonal e).pieces (N.polygonal f).pieces |
| simpa only [polygonIntersection_eq_fin_sum] using hsum |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The cycle engine computes the signed intersection of the original, arbitrary |
| normalized subdivision, independently of the minimal-chain presentation. -/ |
| noncomputable section |
| open Set Zarankiewicz |
| namespace Paper170Geometry.VertexTruncation |
| variable {Vertex Edge : Type} [Fintype Vertex] {graph : Paper170.Graph Vertex Edge} |
|
|
| theorem signedIntersection_eq_interiorIntersection (N : Paper170.NormalizedDrawing graph) |
| (V : VertexTruncation N.toContinuousDrawing) (e f : Edge) : |
| V.signedIntersection e f = Paper170.interiorIntersection N e f := by |
| classical |
| by_cases hef : e=f |
| · subst f |
| rw [V.signedIntersection_self] |
| simp [Paper170.interiorIntersection] |
| · apply signedIntersection_eq_of_matching_signs N V e f hef |
| intro i j k l x hxi hxj hxk hxl |
| have hxe := V.range_subset_edgeInterior e (V.piece_subset_range e i.val i.isLt |
| (openSegmentTrace_subset_segment _ _ hxi)) |
| have hxf := V.range_subset_edgeInterior f (V.piece_subset_range f j.val j.isLt |
| (openSegmentTrace_subset_segment _ _ hxj)) |
| have hx : x∈Paper170.crossingPoints N.toContinuousDrawing := ⟨e,f,hef,hxe,hxf⟩ |
| obtain ⟨r,hr,he⟩ := truncated_direction_pos_mul_original N V e i k x hx hxi hxk |
| obtain ⟨s,hs,hf⟩ := truncated_direction_pos_mul_original N V f j l x hx hxj hxl |
| exact segmentCrossingSign_eq_of_pos_directions _ _ _ _ _ _ _ _ x r s hr hs he hf hxi hxj hxk hxl |
|
|
| end Paper170Geometry.VertexTruncation |
|
|
| end |
| end |
|
|
| end OAI |
|
|
