| import OAI.Combinatorics.CompleteCrossing.Tracks |
| import OAI.Combinatorics.Crossing.MovingDiskClearance |
| import OAI.Combinatorics.Crossing.NormalOffsetExterior |
|
|
| namespace OAI |
|
|
| /-! Radial track displacements and connector clearance from other disks and tracks. -/ |
|
|
| section |
|
|
| /-! Endpoint radial directions have the correct sign in the fixed left-to-right edge orientation. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry |
|
|
| theorem dot2_scaled_self (v : Plane) (a b : ℝ) : dot2 (a • v) (b • v)=a*b*squaredLength v := by |
| dsimp [dot2,squaredLength] |
| ring |
|
|
| namespace VertexTruncation |
| variable {D : Paper170.ContinuousDrawing graph} (V : VertexTruncation D) |
|
|
| theorem start_radial_dot_pos (e : Edge) : |
| 0 < dot2 (V.startPort e-D.vertex ((graph.src e))) (trackDirection (V.vertices e) 0) := by |
| have hp := V.pieces_pos e |
| have hα := (radialPort_parameter_half ((V.chain e).vertices 0) ((V.chain e).vertices 1) |
| V.radius V.radius_pos (V.short e).1).1 |
| have hβ := (V.chain e).truncatedScale_pos V.radius V.radius_pos (V.short e).1 (V.short e).2 0 |
| have he : V.startPort e-D.vertex ((graph.src e))= |
| (V.chain e).startFraction V.radius • trackDirection (V.chain e).vertices 0 := by |
| have hh := radialPort_sub ((V.chain e).vertices 0) ((V.chain e).vertices 1) V.radius |
| change (V.chain e).startPort V.radius-(V.chain e).vertices 0= |
| (V.chain e).startFraction V.radius • trackDirection (V.chain e).vertices 0 at hh |
| simpa only [startPort,(V.chain e).first] using hh |
| rw [he] |
| change 0 < dot2 _ (trackDirection ((V.chain e).truncatedVertices V.radius) 0) |
| rw [(V.chain e).truncated_direction V.radius 0 hp,dot2_scaled_self] |
| exact mul_pos (mul_pos hα hβ) |
| (squaredLength_pos _ ((V.chain e).minimum_piece_ne_zero (V.minimal e) 0 hp)) |
|
|
| theorem end_radial_dot_neg (e : Edge) : |
| dot2 (V.endPort e-D.vertex ((graph.dst e))) |
| (trackDirection (V.vertices e) (V.pieces e-1)) < 0 := by |
| have hp := V.pieces_pos e |
| have hp' : 0 < (V.chain e).pieces := hp |
| have hi : V.pieces e-1 < V.pieces e := by omega |
| have hα := (radialPort_parameter_half ((V.chain e).vertices ((V.chain e).pieces-1)) |
| ((V.chain e).vertices (V.chain e).pieces) V.radius V.radius_pos (V.short e).2).1 |
| have hβ := (V.chain e).truncatedScale_pos V.radius V.radius_pos (V.short e).1 (V.short e).2 (V.pieces e-1) |
| have he : V.endPort e-D.vertex ((graph.dst e))= |
| (-(V.chain e).endFraction V.radius) • trackDirection (V.chain e).vertices (V.pieces e-1) := by |
| have hh : (V.chain e).endPort V.radius-(V.chain e).vertices (V.chain e).pieces= |
| (-(V.chain e).endFraction V.radius) • trackDirection (V.chain e).vertices (V.pieces e-1) := by |
| rw [(V.chain e).endPort_eq_fraction] |
| simp only [trackDirection,pieces,show (V.chain e).pieces-1+1=(V.chain e).pieces by omega,neg_smul] |
| abel |
| simpa only [endPort,(V.chain e).last] using hh |
| rw [he] |
| change dot2 _ (trackDirection ((V.chain e).truncatedVertices V.radius) (V.pieces e-1)) < 0 |
| rw [(V.chain e).truncated_direction V.radius _ hi,dot2_scaled_self] |
| exact mul_neg_of_neg_of_pos (mul_neg_of_neg_of_pos (neg_neg_of_pos hα) hβ) |
| (squaredLength_pos _ ((V.chain e).minimum_piece_ne_zero (V.minimal e) _ hi)) |
|
|
| theorem bend_ne_zero (e : Edge) (i : ℕ) (hi0 : 0 < i) (hin : i < V.pieces e) : |
| det2 (trackDirection (V.vertices e) (i-1)) (trackDirection (V.vertices e) i) ≠ 0 := by |
| exact (V.chain e).truncated_bend_ne_zero (V.minimal e) V.radius V.radius_pos |
| (V.short e).1 (V.short e).2 i hi0 hin |
|
|
| theorem exists_forward_radius (e : Edge) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, |ε|<δ → ∀ i : Fin (V.pieces e), |
| ∃ ℓ : ℝ, 0<ℓ ∧ shiftedTrack (V.vertices e) (V.pieces e) ε (i.val+1)- |
| shiftedTrack (V.vertices e) (V.pieces e) ε i.val=ℓ • trackDirection (V.vertices e) i.val := |
| exists_forward_shiftedTrack_radius (V.vertices e) (V.pieces e) (V.direction_ne_zero e) (V.bend_ne_zero e) |
|
|
| end VertexTruncation |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The actual first and last shifted segments remain outside their incident vertex disks. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.VertexTruncation |
| variable {D : Paper170.ContinuousDrawing graph} (V : VertexTruncation D) |
|
|
| theorem shifted_first_piece_exterior (e : Edge) (ε ℓ : ℝ) |
| (hε : ε ≠ 0) (hℓ : 0<ℓ) |
| (hdir : shiftedTrack (V.vertices e) (V.pieces e) ε 1- |
| shiftedTrack (V.vertices e) (V.pieces e) ε 0=ℓ • trackDirection (V.vertices e) 0) : |
| segment ℝ (shiftedTrack (V.vertices e) (V.pieces e) ε 0) |
| (shiftedTrack (V.vertices e) (V.pieces e) ε 1) ⊆ |
| {x | V.radius^2 < squaredLength (x-D.vertex ((graph.src e)))} := by |
| have hstart : shiftedTrack (V.vertices e) (V.pieces e) ε 0= |
| V.startPort e+ε • normalShift (trackDirection (V.vertices e) 0) := by |
| simp [shiftedTrack,trackOffset,V.vertices_first] |
| have hc : squaredLength (V.startPort e-D.vertex ((graph.src e)))=V.radius^2 := by |
| simpa only [V.spokePort_left] using V.spokePort_circle ((graph.src e)) (V.leftIncident e) |
| have hp : det2 (trackDirection (V.vertices e) 0) (V.startPort e-D.vertex ((graph.src e)))=0 := by |
| simpa [spokeDirection,spokeIndex,leftIncident,spokePort,drawingSpokePort,startPort] using |
| V.spokeDirection_parallel ((graph.src e)) (V.leftIncident e) |
| rw [hstart] at hdir ⊢ |
| exact normal_offset_segment_exterior _ _ _ _ _ ε ℓ (V.direction_ne_zero e 0 (V.pieces_pos e)) |
| hc hp (mul_nonneg hℓ.le (V.start_radial_dot_pos e).le) hε hdir |
|
|
| theorem shifted_last_piece_exterior (e : Edge) (ε ℓ : ℝ) |
| (hε : ε ≠ 0) (hℓ : 0<ℓ) |
| (hdir : shiftedTrack (V.vertices e) (V.pieces e) ε (V.pieces e)- |
| shiftedTrack (V.vertices e) (V.pieces e) ε (V.pieces e-1)= |
| ℓ • trackDirection (V.vertices e) (V.pieces e-1)) : |
| segment ℝ (shiftedTrack (V.vertices e) (V.pieces e) ε (V.pieces e-1)) |
| (shiftedTrack (V.vertices e) (V.pieces e) ε (V.pieces e)) ⊆ |
| {x | V.radius^2 < squaredLength (x-D.vertex ((graph.dst e)))} := by |
| have hn : V.pieces e ≠ 0 := ne_of_gt (V.pieces_pos e) |
| have hi : V.pieces e-1 < V.pieces e := by have hh := V.pieces_pos e; omega |
| have hend : shiftedTrack (V.vertices e) (V.pieces e) ε (V.pieces e)= |
| V.endPort e+ε • normalShift (trackDirection (V.vertices e) (V.pieces e-1)) := by |
| simp [shiftedTrack,trackOffset,hn,V.vertices_last] |
| have hc : squaredLength (V.endPort e-D.vertex ((graph.dst e)))=V.radius^2 := by |
| simpa only [V.spokePort_right] using V.spokePort_circle ((graph.dst e)) (V.rightIncident e) |
| have hp : det2 (trackDirection (V.vertices e) (V.pieces e-1)) (V.endPort e-D.vertex ((graph.dst e)))=0 := by |
| simpa [spokeDirection,spokeIndex,rightIncident,spokePort,drawingSpokePort,endPort,(graph.loopless e).symm] using |
| V.spokeDirection_parallel ((graph.dst e)) (V.rightIncident e) |
| have hrev : shiftedTrack (V.vertices e) (V.pieces e) ε (V.pieces e-1)- |
| (V.endPort e+ε • normalShift (trackDirection (V.vertices e) (V.pieces e-1)))= |
| (-ℓ) • trackDirection (V.vertices e) (V.pieces e-1) := by |
| rw [neg_smul,← hdir,hend] |
| abel |
| rw [segment_symm,hend] |
| exact normal_offset_segment_exterior _ _ _ _ _ ε (-ℓ) (V.direction_ne_zero e _ hi) |
| hc hp (mul_nonneg_of_nonpos_of_nonpos (by linarith) (V.end_radial_dot_neg e).le) hε hrev |
|
|
| end Paper170Geometry.VertexTruncation |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual connectors stay on the displaced supporting line and avoid their own original spoke piece. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry |
|
|
| theorem det2_segmentPoint_sub (d p a b : Plane) (t : ℝ) : |
| det2 d (segmentPoint a b t-p)=(1-t)*det2 d (a-p)+t*det2 d (b-p) := by |
| dsimp [det2,segmentPoint] |
| ring |
|
|
| theorem segment_line_level_zero (a b p : Plane) (hp : p=a ∨ p=b) (t : ℝ) : |
| det2 (b-a) (segmentPoint a b t-p)=0 := by |
| rcases hp with rfl | rfl <;> dsimp [det2,segmentPoint] <;> ring |
| |
| namespace VertexTruncation |
| variable {D : Paper170.ContinuousDrawing graph} (V : VertexTruncation D) |
|
|
| theorem spokePort_is_piece_endpoint (j : Vertex) (e : IncidentEdges graph j) : |
| V.spokePort j e=V.vertices e.val (V.spokeIndex j e) ∨ |
| V.spokePort j e=V.vertices e.val (V.spokeIndex j e+1) := by |
| unfold spokePort drawingSpokePort spokeIndex |
| split_ifs |
| · exact Or.inl (V.vertices_first e.val).symm |
| · right |
| have hp := V.pieces_pos e.val |
| rw [show V.pieces e.val-1+1=V.pieces e.val by omega] |
| exact (V.vertices_last e.val).symm |
|
|
| end VertexTruncation |
| namespace TruncatedSpokeCoordinates |
| variable {D : Paper170.ContinuousDrawing graph} {V : VertexTruncation D} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem continuous_tangentPort (j : Vertex) (e : IncidentEdges graph j) : |
| Continuous (C.tangentPort j e) := by |
| change Continuous (fun ε : ℝ => V.spokePort j e+ε • normalShift (V.spokeDirection j e)) |
| fun_prop |
|
|
| theorem tangentPort_zero (j : Vertex) (e : IncidentEdges graph j) : |
| C.tangentPort j e 0=V.spokePort j e := by simp [tangentPort] |
|
|
| theorem tangentPort_level (j : Vertex) (e : IncidentEdges graph j) (ε : ℝ) : |
| det2 (V.spokeDirection j e) (C.tangentPort j e ε-V.spokePort j e)=ε := by |
| simp only [tangentPort,add_sub_cancel_left,det2_smul_right, |
| det2_normalShift _ (V.spokeDirection_ne_zero j e),mul_one] |
|
|
| theorem connector_level (j : Vertex) (e : IncidentEdges graph j) (ε t : ℝ) : |
| det2 (V.spokeDirection j e) |
| (segmentPoint (C.shiftedPort j e ε) (C.tangentPort j e ε) t-V.spokePort j e)=ε := by |
| rw [det2_segmentPoint_sub,C.shiftedPort_level,C.tangentPort_level] |
| ring |
|
|
| theorem connector_avoids_incident_piece (j : Vertex) (e : IncidentEdges graph j) |
| (ε : ℝ) (hε : ε ≠ 0) : |
| ¬ ClosedSegmentsMeet (V.vertices e.val (V.spokeIndex j e)) |
| (V.vertices e.val (V.spokeIndex j e+1)) (C.shiftedPort j e ε) (C.tangentPort j e ε) := by |
| rintro ⟨s,hs,t,ht,he⟩ |
| have hzero := segment_line_level_zero (V.vertices e.val (V.spokeIndex j e)) |
| (V.vertices e.val (V.spokeIndex j e+1)) (V.spokePort j e) (V.spokePort_is_piece_endpoint j e) s |
| change det2 (V.spokeDirection j e) |
| (segmentPoint (V.vertices e.val (V.spokeIndex j e)) (V.vertices e.val (V.spokeIndex j e+1)) s-V.spokePort j e)=0 at hzero |
| rw [he,C.connector_level] at hzero |
| exact hε hzero |
|
|
| end TruncatedSpokeCoordinates |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! One genuine radius keeps every shifted exterior edge track outside every vertex disk. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.VertexTruncation |
| variable {D : Paper170.ContinuousDrawing graph} (V : VertexTruncation D) |
|
|
| theorem exists_nonincident_shift_radius [Fintype Vertex] [Fintype Edge] : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, |ε|<δ → ∀ j e (i : Fin (V.pieces e)), |
| ¬ ((j=(graph.src e) ∧ i.val=0) ∨ (j=(graph.dst e) ∧ i.val+1=V.pieces e)) → |
| segment ℝ (shiftedTrack (V.vertices e) (V.pieces e) ε i.val) |
| (shiftedTrack (V.vertices e) (V.pieces e) ε (i.val+1)) ⊆ |
| {x | V.radius^2 < squaredLength (x-D.vertex j)} := by |
| classical |
| let Bad := Σ j : Vertex, Σ e : Edge, |
| {i : Fin (V.pieces e) // ¬ ((j=(graph.src e) ∧ i.val=0) ∨ (j=(graph.dst e) ∧ i.val+1=V.pieces e))} |
| have hdis (b : Bad) : Disjoint (closedRoundDisk (D.vertex b.1) V.radius) |
| (segment ℝ (V.vertices b.2.1 b.2.2.val.val) (V.vertices b.2.1 (b.2.2.val.val+1))) := by |
| exact (V.piece_clearance b.1 b.2.1 b.2.2.val b.2.2.property).mono_right |
| ((V.chain b.2.1).truncated_piece_subset V.radius V.radius_pos |
| (V.weak_short b.2.1).1 (V.weak_short b.2.1).2 _ b.2.2.val.isLt) |
| have hex (b : Bad) := exists_shifted_segment_disk_radius |
| (V.vertices b.2.1 b.2.2.val.val) (V.vertices b.2.1 (b.2.2.val.val+1)) |
| (trackOffset (V.vertices b.2.1) (V.pieces b.2.1) b.2.2.val.val) |
| (trackOffset (V.vertices b.2.1) (V.pieces b.2.1) (b.2.2.val.val+1)) |
| (D.vertex b.1) V.radius (hdis b) |
| choose radius hpos hprop using hex |
| obtain ⟨δ,hδ,hbound⟩ := finite_positive_lower_bound Finset.univ radius (fun b _ => hpos b) |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε j e i hi |
| exact hprop ⟨j,e,⟨i,hi⟩⟩ ε (hε.trans (hbound ⟨j,e,⟨i,hi⟩⟩ (Finset.mem_univ _))) |
|
|
| theorem exists_all_vertex_disk_radius [Fintype Vertex] [Fintype Edge] : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε|<δ → ∀ j e (i : Fin (V.pieces e)), |
| segment ℝ (shiftedTrack (V.vertices e) (V.pieces e) ε i.val) |
| (shiftedTrack (V.vertices e) (V.pieces e) ε (i.val+1)) ⊆ |
| {x | V.radius^2 < squaredLength (x-D.vertex j)} := by |
| classical |
| obtain ⟨δ₀,hδ₀,hclear⟩ := V.exists_nonincident_shift_radius |
| choose radius hpos hforward using V.exists_forward_radius |
| obtain ⟨δ₁,hδ₁,hbound⟩ := finite_positive_lower_bound Finset.univ radius (fun e _ => hpos e) |
| refine ⟨min δ₀ δ₁,lt_min hδ₀ hδ₁,?_⟩ |
| intro ε hε hsmall j e i |
| obtain ⟨ℓ,hℓ,hdir⟩ := hforward e ε |
| (hsmall.trans_le (min_le_right _ _ ) |>.trans (hbound e (Finset.mem_univ _))) i |
| by_cases hleft : j=(graph.src e) ∧ i.val=0 |
| · rcases hleft with ⟨rfl,hi⟩ |
| simpa only [hi,zero_add] using V.shifted_first_piece_exterior e ε ℓ hε hℓ (by simpa only [hi,zero_add] using hdir) |
| · by_cases hright : j=(graph.dst e) ∧ i.val+1=V.pieces e |
| · rcases hright with ⟨rfl,hi⟩ |
| have hi' : i.val=V.pieces e-1 := by omega |
| have hn : V.pieces e-1+1=V.pieces e := by omega |
| simpa only [hi',hn] using V.shifted_last_piece_exterior e ε ℓ hε hℓ (by simpa only [hi',hn] using hdir) |
| · exact hclear ε (hsmall.trans_le (min_le_left _ _)) j e i (not_or.mpr ⟨hleft,hright⟩) |
|
|
| end Paper170Geometry.VertexTruncation |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Every shrinking circle connector stays outside all other vertex disks at one common radius. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| open scoped Topology |
| variable {D : Paper170.ContinuousDrawing graph} {V : VertexTruncation D} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem exists_connector_other_disks_radius [Fintype Vertex] [Fintype Edge] : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, |ε|<δ → ∀ j k, j ≠ k → ∀ f : IncidentEdges graph j, |
| segment ℝ (C.shiftedPort j f ε) (C.tangentPort j f ε) ⊆ |
| {x | V.radius^2 < squaredLength (x-D.vertex k)} := by |
| classical |
| let Pair := Σ j : Vertex, Σ k : {k : Vertex // j ≠ k}, IncidentEdges graph j |
| have hstart (b : Pair) : segment ℝ (C.shiftedPort b.1 b.2.2 0) (C.tangentPort b.1 b.2.2 0) ⊆ |
| {x | V.radius^2 < squaredLength (x-D.vertex b.2.1.val)} := by |
| rw [C.shiftedPort_zero,C.tangentPort_zero,segment_same] |
| intro x hx |
| have he : x=V.spokePort b.1 b.2.2 := hx |
| subst x |
| change V.radius^2 < squaredLength (V.spokePort b.1 b.2.2-D.vertex b.2.1.val) |
| by_contra hh |
| have hj : V.spokePort b.1 b.2.2 ∈ closedRoundDisk (D.vertex b.1) V.radius := by |
| change squaredLength (V.spokePort b.1 b.2.2-D.vertex b.1) ≤ V.radius^2 |
| rw [V.spokePort_circle] |
| have hk : V.spokePort b.1 b.2.2 ∈ closedRoundDisk (D.vertex b.2.1.val) V.radius := not_lt.mp hh |
| exact Set.disjoint_left.mp (V.disks_disjoint b.1 b.2.1.val b.2.1.property) hj hk |
| have hevent (b : Pair) := eventually_moving_segment_outside_disk |
| (C.shiftedPort b.1 b.2.2) (C.tangentPort b.1 b.2.2) |
| (C.continuous_shiftedPort b.1 b.2.2) (C.continuous_tangentPort b.1 b.2.2) |
| (D.vertex b.2.1.val) V.radius (hstart b) |
| have hex (b : Pair) := Metric.eventually_nhds_iff.mp (hevent b) |
| choose radius hpos hprop using hex |
| obtain ⟨δ,hδ,hbound⟩ := finite_positive_lower_bound Finset.univ radius (fun b _ => hpos b) |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε j k hjk f |
| exact hprop ⟨j,⟨k,hjk⟩,f⟩ (by simpa [Real.dist_eq] using hε.trans (hbound ⟨j,⟨k,hjk⟩,f⟩ (Finset.mem_univ _))) |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! One two-sided physical radius excludes every connector/track meeting, including shared edges. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| open scoped Topology |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem exists_nonincident_connector_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, |ε|<δ → ∀ j (f : IncidentEdges graph j) e (i : Fin (V.pieces e)), |
| ¬ (e=f.val ∧ i.val=V.spokeIndex j f) → |
| ¬ ClosedSegmentsMeet (V.vertices e i.val) (V.vertices e (i.val+1)) |
| (C.shiftedPort j f ε) (C.tangentPort j f ε) := by |
| classical |
| let Bad := Σ j : Vertex, Σ f : IncidentEdges graph j, Σ e : Edge, |
| {i : Fin (V.pieces e) // ¬ (e=f.val ∧ i.val=V.spokeIndex j f)} |
| have hnot (b : Bad) : ¬ ClosedSegmentsMeet |
| (V.vertices b.2.2.1 b.2.2.2.val.val) (V.vertices b.2.2.1 (b.2.2.2.val.val+1)) |
| (V.spokePort b.1 b.2.1) (V.spokePort b.1 b.2.1) := by |
| rintro ⟨s,hs,t,ht,he⟩ |
| have he0 : segmentPoint (V.spokePort b.1 b.2.1) (V.spokePort b.1 b.2.1) t=V.spokePort b.1 b.2.1 := by |
| unfold segmentPoint |
| rw [← add_smul] |
| simp |
| rw [he0] at he |
| have hx : V.spokePort b.1 b.2.1 ∈ |
| segment ℝ (V.vertices b.2.2.1 b.2.2.2.val.val) (V.vertices b.2.2.1 (b.2.2.2.val.val+1)) := by |
| rw [← Path.range_segment] |
| refine ⟨⟨s,hs⟩,?_⟩ |
| simpa only [Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using he |
| exact b.2.2.2.property (V.spokePort_piece_unique hnorm b.1 b.2.1 b.2.2.1 b.2.2.2.val hx) |
| have hevent (b : Bad) := eventually_moving_disjoint_segments |
| (V.vertices b.2.2.1 b.2.2.2.val.val) (V.vertices b.2.2.1 (b.2.2.2.val.val+1)) |
| (V.spokePort b.1 b.2.1) (V.spokePort b.1 b.2.1) (C.shiftedPort b.1 b.2.1) (C.tangentPort b.1 b.2.1) |
| (C.continuous_shiftedPort b.1 b.2.1).continuousAt (C.continuous_tangentPort b.1 b.2.1).continuousAt |
| (C.shiftedPort_zero b.1 b.2.1) (C.tangentPort_zero b.1 b.2.1) (hnot b) |
| have hex (b : Bad) := Metric.eventually_nhds_iff.mp (hevent b) |
| choose radius hpos hprop using hex |
| obtain ⟨δ,hδ,hbound⟩ := finite_positive_lower_bound Finset.univ radius (fun b _ => hpos b) |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε j f e i hi |
| exact hprop ⟨j,f,e,⟨i,hi⟩⟩ (by simpa [Real.dist_eq] using hε.trans (hbound ⟨j,f,e,⟨i,hi⟩⟩ (Finset.mem_univ _))) |
|
|
| theorem exists_all_connector_track_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε|<δ → ∀ j (f : IncidentEdges graph j) e (i : Fin (V.pieces e)), |
| ¬ ClosedSegmentsMeet (V.vertices e i.val) (V.vertices e (i.val+1)) |
| (C.shiftedPort j f ε) (C.tangentPort j f ε) := by |
| obtain ⟨δ,hδ,hh⟩ := C.exists_nonincident_connector_radius hnorm |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε hsmall j f e i |
| by_cases hi : e=f.val ∧ i.val=V.spokeIndex j f |
| · rcases hi with ⟨rfl,hi⟩ |
| simpa only [hi] using C.connector_avoids_incident_piece j f ε hε |
| · exact hh ε hsmall j f e i hi |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| end OAI |
|
|
