| import OAI.Combinatorics.CompleteCrossing.RadialTracks |
| import OAI.Combinatorics.Crossing.DiskSegments |
| import OAI.Combinatorics.Crossing.FiniteScaledRadius |
| import OAI.Combinatorics.Crossing.RoundCircleChords |
| import OAI.Combinatorics.Crossing.RoundDiskChords |
| import OAI.Combinatorics.Crossing.SegmentTraversalReversal |
|
|
| namespace OAI |
|
|
| /-! Uniform spoke radii, vertex chords, and separation from exterior tracks. -/ |
|
|
| section |
|
|
| /-! Common physical radii for every vertex of the actual drawing, fixed before any cycle choices. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| variable {D : Paper170.ContinuousDrawing graph} {V : VertexTruncation D} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem port_injective (C : TruncatedSpokeCoordinates V) (j : Vertex) : Function.Injective (V.spokePort j) := by |
| intro e f he |
| apply C.node_injective j |
| apply circlePoint_injective |
| apply similarityPoint_injective (D.vertex j) (C.frame j) (C.frame_ne_zero j) |
| simpa only [C.point_eq] using he |
|
|
| theorem exists_uniform_endpoint_radius [Fintype Vertex] [Fintype Edge] : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε|<δ → ∀ j : Vertex, |
| (∀ e f : IncidentEdges graph j, V.spokePort j e ≠ C.shiftedPort j f ε) ∧ |
| Function.Injective (fun e : IncidentEdges graph j => C.shiftedPort j e ε) := by |
| classical |
| choose radius hpos hprop using C.exists_endpoint_separation_radius |
| obtain ⟨δ,hδ,hbound⟩ := finite_positive_lower_bound Finset.univ radius (fun j _ => hpos j) |
| exact ⟨δ,hδ,fun ε hε hsmall j => hprop j ε hε (hsmall.trans (hbound j (Finset.mem_univ _)))⟩ |
|
|
| theorem exists_uniform_scale_radius [Fintype Vertex] [Fintype Edge] (_C : TruncatedSpokeCoordinates V) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, |ε|<δ → ∀ j (e : IncidentEdges graph j), |V.spokeScale j e*ε|<1 := by |
| let Spoke := Σ j : Vertex, IncidentEdges graph j |
| obtain ⟨δ,hδ,hbound⟩ := finite_scaled_radius |
| (fun s : Spoke => V.spokeScale s.1 s.2) (fun _ => (1 : ℝ)) (by intro s; norm_num) |
| exact ⟨δ,hδ,fun ε hε j e => hbound ε hε ⟨j,e⟩⟩ |
|
|
| theorem exists_uniform_average_radius [Fintype Vertex] [Fintype Edge] : |
| ∃ δ : ℝ, 0<δ ∧ ∀ t : ℝ, 0<t → t<δ → ∀ j (a b c d : IncidentEdges graph j), |
| a ≠ b → c ≠ d → |
| (segmentCrossingSign (V.spokePort j a) (V.spokePort j b) (C.shiftedPort j c t) (C.shiftedPort j d t)+ |
| segmentCrossingSign (V.spokePort j a) (V.spokePort j b) (C.shiftedPort j c (-t)) (C.shiftedPort j d (-t)))/2 = |
| chordEndpointPairing (C.node j a) (C.node j b) (C.node j c) (C.node j d) := by |
| classical |
| choose radius hpos hprop using C.exists_vertex_average_radius |
| obtain ⟨δ,hδ,hbound⟩ := finite_positive_lower_bound Finset.univ radius (fun j _ => hpos j) |
| exact ⟨δ,hδ,fun t ht hsmall j => hprop j t ht (hsmall.trans (hbound j (Finset.mem_univ _)))⟩ |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual local chords have transverse meetings and avoid the exterior connector seams. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| variable {D : Paper170.ContinuousDrawing graph} {V : VertexTruncation D} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem shifted_chords_transverse (j : Vertex) (a b c d : IncidentEdges graph j) |
| (ε : ℝ) (hsmall : ∀ e : IncidentEdges graph j, |V.spokeScale j e*ε|≤1) |
| (hsep : ∀ e f : IncidentEdges graph j, V.spokePort j e ≠ C.shiftedPort j f ε) |
| (hinj : Function.Injective (fun e : IncidentEdges graph j => C.shiftedPort j e ε)) |
| (hab : a ≠ b) (hcd : c ≠ d) : |
| SegmentPairTransverse (V.spokePort j a) (V.spokePort j b) (C.shiftedPort j c ε) (C.shiftedPort j d ε) := |
| round_circle_chords_transverse (D.vertex j) _ _ _ _ V.radius |
| (V.spokePort_circle j a) (V.spokePort_circle j b) |
| (C.shiftedPort_circle j c ε (hsmall c)) (C.shiftedPort_circle j d ε (hsmall d)) |
| ((C.port_injective j).ne hab) (hinj.ne hcd) (hsep a c) (hsep a d) (hsep b c) (hsep b d) |
|
|
| theorem chord_connector_disjoint (j : Vertex) (a b e : IncidentEdges graph j) |
| (ε : ℝ) (hne : ε ≠ 0) (hsmall : |V.spokeScale j e*ε|<1) (hab : a ≠ b) |
| (ha : V.spokePort j a ≠ C.shiftedPort j e ε) (hb : V.spokePort j b ≠ C.shiftedPort j e ε) : |
| ¬ ClosedSegmentsMeet (V.spokePort j a) (V.spokePort j b) (C.shiftedPort j e ε) (C.tangentPort j e ε) := by |
| have hunit (f : IncidentEdges graph j) : squaredLength (circlePoint (C.node j f))=1 := circlePoint_on_unit_circle _ |
| have haq : circlePoint (C.node j a) ≠ circleOffset (circlePoint (C.node j e)) (V.spokeScale j e*ε) := by |
| intro he |
| apply ha |
| rw [← C.point_eq j a] |
| exact congrArg (similarityPoint (D.vertex j) (C.frame j)) he |
| have hbq : circlePoint (C.node j b) ≠ circleOffset (circlePoint (C.node j e)) (V.spokeScale j e*ε) := by |
| intro he |
| apply hb |
| rw [← C.point_eq j b] |
| exact congrArg (similarityPoint (D.vertex j) (C.frame j)) he |
| have hd := unit_chord_connector_disjoint (circlePoint (C.node j a)) (circlePoint (C.node j b)) |
| (circlePoint (C.node j e)) (hunit a) (hunit b) (hunit e) |
| (fun he => hab (C.node_injective j (circlePoint_injective he))) |
| (V.spokeScale j e*ε) hsmall (mul_ne_zero (V.spokeScale_ne_zero j e) hne) haq hbq |
| rintro ⟨s,hs,t,ht,he⟩ |
| apply hd |
| refine ⟨s,hs,t,ht,?_⟩ |
| apply similarityPoint_injective (D.vertex j) (C.frame j) (C.frame_ne_zero j) |
| simpa only [similarityPoint_segment,C.point_eq,← C.tangentPort_eq,shiftedPort] using he |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Original exterior tracks avoid every shifted vertex chord, including at boundary seams. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry |
| namespace VertexTruncation |
| variable {D : Paper170.ContinuousDrawing graph} (V : VertexTruncation D) |
|
|
| theorem track_in_disk_is_port (j : Vertex) (e : Edge) (x : Plane) |
| (hx : x ∈ Set.range (V.path e)) (hmem : x ∈ closedRoundDisk (D.vertex j) V.radius) : |
| ∃ f : IncidentEdges graph j, x=V.spokePort j f := by |
| rcases V.vertex_disk_cases j e x hx with ⟨hj,he⟩ | ⟨hj,he⟩ | hout |
| · refine ⟨⟨e,Or.inl hj⟩,?_⟩ |
| simpa only [spokePort,drawingSpokePort,ite_eq_left hj,startPort] using he |
| · have hnot : j ≠ (graph.src e) := by |
| intro hh |
| exact (graph.loopless e) (hh.symm.trans hj) |
| refine ⟨⟨e,Or.inr hj⟩,?_⟩ |
| simpa only [spokePort,drawingSpokePort,ite_eq_right hnot,endPort] using he |
| · exact False.elim (not_lt_of_ge hmem hout) |
|
|
| end VertexTruncation |
| namespace TruncatedSpokeCoordinates |
| variable {D : Paper170.ContinuousDrawing graph} {V : VertexTruncation D} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem track_shifted_chord_disjoint (j : Vertex) (c d : IncidentEdges graph j) |
| (ε : ℝ) (hc : |V.spokeScale j c*ε|≤1) (hd : |V.spokeScale j d*ε|≤1) |
| (hsep : ∀ f g : IncidentEdges graph j, V.spokePort j f ≠ C.shiftedPort j g ε) |
| (e : Edge) (i : Fin (V.pieces e)) : |
| ¬ ClosedSegmentsMeet (V.vertices e i.val) (V.vertices e (i.val+1)) |
| (C.shiftedPort j c ε) (C.shiftedPort j d ε) := by |
| intro hmeet |
| obtain ⟨x,hx,hy⟩ := (closedSegmentsMeet_iff_nonempty_inter _ _ _ _).mp hmeet |
| have hcirclec := C.shiftedPort_circle j c ε hc |
| have hcircled := C.shiftedPort_circle j d ε hd |
| have hmem := round_chord_subset_closed_disk (D.vertex j) _ _ V.radius hcirclec hcircled hy |
| obtain ⟨f,hf⟩ := V.track_in_disk_is_port j e x (V.piece_subset_range e i.val i.isLt hx) hmem |
| have hboundary : squaredLength (x-D.vertex j)=V.radius^2 := by rw [hf,V.spokePort_circle] |
| rcases round_chord_boundary_endpoints (D.vertex j) _ _ x V.radius hcirclec hcircled hy hboundary with h | h |
| · exact hsep f c (hf.symm.trans h) |
| · exact hsep f d (hf.symm.trans h) |
|
|
| end TruncatedSpokeCoordinates |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! All disk/exterior and different-disk segment interactions are genuinely disjoint. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry |
|
|
| theorem round_chord_exterior_disjoint (z a b c d : Plane) (r : ℝ) |
| (ha : squaredLength (a-z)=r^2) (hb : squaredLength (b-z)=r^2) |
| (hout : segment ℝ c d ⊆ {x | r^2 < squaredLength (x-z)}) : ¬ ClosedSegmentsMeet a b c d := by |
| intro hh |
| obtain ⟨x,hx,hy⟩ := (closedSegmentsMeet_iff_nonempty_inter _ _ _ _).mp hh |
| have hle := round_chord_subset_closed_disk z a b r ha hb hx |
| have hlt : r^2 < squaredLength (x-z) := hout hy |
| change squaredLength (x-z) ≤ r^2 at hle |
| exact not_lt_of_ge hle hlt |
|
|
| namespace TruncatedSpokeCoordinates |
| variable {D : Paper170.ContinuousDrawing graph} {V : VertexTruncation D} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem different_vertex_chords_disjoint (j k : Vertex) (hjk : j ≠ k) |
| (a b : IncidentEdges graph j) (c d : IncidentEdges graph k) (ε : ℝ) |
| (hc : |V.spokeScale k c*ε|≤1) (hd : |V.spokeScale k d*ε|≤1) : |
| ¬ ClosedSegmentsMeet (V.spokePort j a) (V.spokePort j b) (C.shiftedPort k c ε) (C.shiftedPort k d ε) := by |
| apply not_closedSegmentsMeet_of_disjoint |
| exact (V.disks_disjoint j k hjk).mono |
| (round_chord_subset_closed_disk _ _ _ _ (V.spokePort_circle j a) (V.spokePort_circle j b)) |
| (round_chord_subset_closed_disk _ _ _ _ (C.shiftedPort_circle k c ε hc) (C.shiftedPort_circle k d ε hd)) |
|
|
| theorem chord_connector_disjoint_all_passages (j : Vertex) (a b f : IncidentEdges graph j) |
| (ε : ℝ) (hne : ε ≠ 0) (hsmall : |V.spokeScale j f*ε|<1) |
| (ha : V.spokePort j a ≠ C.shiftedPort j f ε) (hb : V.spokePort j b ≠ C.shiftedPort j f ε) : |
| ¬ ClosedSegmentsMeet (V.spokePort j a) (V.spokePort j b) (C.shiftedPort j f ε) (C.tangentPort j f ε) := by |
| rintro ⟨s,hs,t,ht,he⟩ |
| have hx : segmentPoint (V.spokePort j a) (V.spokePort j b) s ∈ segment ℝ (V.spokePort j a) (V.spokePort j b) := by |
| rw [← Path.range_segment] |
| exact ⟨⟨s,hs⟩,by simp only [Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint]⟩ |
| have hinside := round_chord_subset_closed_disk (D.vertex j) _ _ V.radius |
| (V.spokePort_circle j a) (V.spokePort_circle j b) hx |
| by_cases htp : 0<t |
| · have hout := C.connector_strictly_outside j f ε hsmall hne t htp |
| rw [← he] at hout |
| exact not_lt_of_ge hinside hout |
| · have ht0 : t=0 := by linarith [ht.1] |
| rw [ht0,segmentPoint_zero] at he |
| have hboundary : squaredLength (segmentPoint (V.spokePort j a) (V.spokePort j b) s-D.vertex j)=V.radius^2 := by |
| rw [he] |
| exact C.shiftedPort_circle j f ε hsmall.le |
| rcases round_chord_boundary_endpoints (D.vertex j) _ _ _ V.radius |
| (V.spokePort_circle j a) (V.spokePort_circle j b) hx hboundary with h | h |
| · exact ha (h.symm.trans he) |
| · exact hb (h.symm.trans he) |
|
|
| end TruncatedSpokeCoordinates |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! One radius verifies every track, connector and chord interaction needed for actual closed-walk assembly. -/ |
| 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_cross_type_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)) ∧ |
| (∀ j (c d : IncidentEdges graph j) e (i : Fin (V.pieces e)), |
| ¬ ClosedSegmentsMeet (V.vertices e i.val) (V.vertices e (i.val+1)) |
| (C.shiftedPort j c ε) (C.shiftedPort j d ε)) ∧ |
| (∀ j (a b : IncidentEdges graph j) e (i : Fin (V.pieces e)), |
| ¬ ClosedSegmentsMeet (V.spokePort j a) (V.spokePort j b) |
| (shiftedTrack (V.vertices e) (V.pieces e) ε i.val) |
| (shiftedTrack (V.vertices e) (V.pieces e) ε (i.val+1))) ∧ |
| (∀ j k (a b : IncidentEdges graph j) (f : IncidentEdges graph k), |
| ¬ ClosedSegmentsMeet (V.spokePort j a) (V.spokePort j b) |
| (C.shiftedPort k f ε) (C.tangentPort k f ε)) ∧ |
| (∀ j k (a b : IncidentEdges graph j) (c d : IncidentEdges graph k), a ≠ b → c ≠ d → |
| SegmentPairTransverse (V.spokePort j a) (V.spokePort j b) (C.shiftedPort k c ε) (C.shiftedPort k d ε)) ∧ |
| (∀ j k, j ≠ k → ∀ (a b : IncidentEdges graph j) (c d : IncidentEdges graph k), |
| ¬ ClosedSegmentsMeet (V.spokePort j a) (V.spokePort j b) (C.shiftedPort k c ε) (C.shiftedPort k d ε)) := by |
| obtain ⟨δ₀,hδ₀,htracks⟩ := V.exists_all_vertex_disk_radius |
| obtain ⟨δ₁,hδ₁,hconnectors⟩ := C.exists_all_connector_track_radius hnorm |
| obtain ⟨δ₂,hδ₂,hother⟩ := C.exists_connector_other_disks_radius |
| obtain ⟨δ₃,hδ₃,hseparate⟩ := C.exists_uniform_endpoint_radius |
| obtain ⟨δ₄,hδ₄,hscale⟩ := C.exists_uniform_scale_radius |
| let δ := min δ₀ (min δ₁ (min δ₂ (min δ₃ δ₄))) |
| have hδ : 0<δ := lt_min hδ₀ (lt_min hδ₁ (lt_min hδ₂ (lt_min hδ₃ hδ₄))) |
| 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 _ _).trans (min_le_left _ _)) |
| have hb₃ : δ≤δ₃ := (min_le_right _ _).trans ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _))) |
| have hb₄ : δ≤δ₄ := (min_le_right _ _).trans ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _))) |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε hsmall |
| have hs := hscale ε (hsmall.trans_le hb₄) |
| have hsep := hseparate ε hε (hsmall.trans_le hb₃) |
| refine ⟨hconnectors ε hε (hsmall.trans_le hb₁),?_,?_,?_,?_,?_⟩ |
| · intro j c d e i |
| exact C.track_shifted_chord_disjoint j c d ε (hs j c).le (hs j d).le (hsep j).1 e i |
| · intro j a b e i |
| exact round_chord_exterior_disjoint (D.vertex j) _ _ _ _ V.radius |
| (V.spokePort_circle j a) (V.spokePort_circle j b) (htracks ε hε (hsmall.trans_le hb₀) j e i) |
| · intro j k a b f |
| by_cases hjk : j=k |
| · subst k |
| exact C.chord_connector_disjoint_all_passages j a b f ε hε (hs j f) ((hsep j).1 a f) ((hsep j).1 b f) |
| · exact round_chord_exterior_disjoint (D.vertex j) _ _ _ _ V.radius |
| (V.spokePort_circle j a) (V.spokePort_circle j b) |
| (hother ε (hsmall.trans_le hb₂) k j (Ne.symm hjk) f) |
| · intro j k a b c d hab hcd |
| by_cases hjk : j=k |
| · subst k |
| exact C.shifted_chords_transverse j a b c d ε (fun e => (hs j e).le) (hsep j).1 (hsep j).2 hab hcd |
| · exact segmentPairTransverse_of_not_meet _ _ _ _ |
| (C.different_vertex_chords_disjoint j k hjk a b c d ε (hs k c).le (hs k d).le) |
| · intro j k hjk a b c d |
| exact C.different_vertex_chords_disjoint j k hjk a b c d ε (hs k c).le (hs k d).le |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| end OAI |
|
|
