| import OAI.Combinatorics.CompleteCrossing.LocalGeometry |
| import OAI.Combinatorics.Crossing.ChainPieceInteriors |
| import OAI.Combinatorics.Crossing.ChainTracePieces |
| import OAI.Combinatorics.Crossing.DrawingSpokePorts |
| import OAI.Combinatorics.Crossing.IncidentSpokePorts |
| import OAI.Combinatorics.Crossing.MinimumChainVertexPieces |
| import OAI.Combinatorics.Crossing.ShortVertexRadius |
| import OAI.Combinatorics.Crossing.VertexPieceClearance |
| import OAI.Combinatorics.Crossing.VertexSpokeBounds |
| import OAI.Combinatorics.Crossing.VertexTraceDisks |
|
|
| namespace OAI |
|
|
| /-! Chain pieces near vertices, incident spoke ports, and short vertex radii. -/ |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem drawing_chain_piece_vertex (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ Q : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) |
| (D.vertex ((graph.dst e))), P.pieces ≤ Q.pieces) |
| (i : ℕ) (hi : i < P.pieces) (j : Vertex) |
| (hx : D.vertex j ∈ segment ℝ (P.vertices i) (P.vertices (i+1))) : |
| (j=(graph.src e) ∧ i=0) ∨ (j=(graph.dst e) ∧ i+1=P.pieces) := by |
| obtain ⟨t,ht⟩ := P.inside i hi hx |
| rcases drawing_edge_vertex_parameter D e t j ht with ⟨_,he⟩ | ⟨_,he⟩ |
| · have hx0 : P.vertices 0 ∈ segment ℝ (P.vertices i) (P.vertices (i+1)) := by |
| simpa only [P.first,he] using hx |
| have hh := P.minimum_vertex_piece_index hmin i 0 hi (by omega) hx0 |
| exact Or.inl ⟨he,by omega⟩ |
| · have hxN : P.vertices P.pieces ∈ segment ℝ (P.vertices i) (P.vertices (i+1)) := by |
| simpa only [P.last,he] using hx |
| have hh := P.minimum_vertex_piece_index hmin i P.pieces hi (by omega) hxN |
| exact Or.inr ⟨he,by omega⟩ |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem drawing_chain_range_eq (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) : |
| Set.range P.path=Set.range (D.edge e) := |
| path_range_eq_of_subset (D.edge e) P.path (D.edge_injective e) P.path_range_subset |
|
|
| theorem drawing_chain_pieces_pos (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) : |
| 0 < P.pieces := by |
| apply P.pieces_pos_of_ne |
| intro h |
| exact graph.loopless e (D.vertex_injective h) |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem drawing_chain_piece_open_subset_interior (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ Q : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) |
| (D.vertex ((graph.dst e))), P.pieces ≤ Q.pieces) |
| (i : ℕ) (hi : i < P.pieces) : |
| openSegmentTrace (P.vertices i) (P.vertices (i+1)) ⊆ Paper170.edgeInterior D e := by |
| intro x hx |
| have hseg := openSegmentTrace_subset_segment _ _ hx |
| have hne := openSegmentTrace_ne_endpoints _ _ |
| (P.minimum_vertices_ne hmin i (i+1) (by omega) (by omega)) x hx |
| apply edgeInterior_of_range_not_vertices D e x (P.inside i hi hseg) |
| intro j hj |
| have hvertex : D.vertex j ∈ segment ℝ (P.vertices i) (P.vertices (i+1)) := hj ▸ hseg |
| rcases drawing_chain_piece_vertex D e P hmin i hi j hvertex with ⟨he,hii⟩ | ⟨he,hii⟩ |
| · apply hne.1 |
| calc |
| x = D.vertex j := hj |
| _ = D.vertex ((graph.src e)) := congrArg D.vertex he |
| _ = P.vertices i := by simpa only [hii] using P.first.symm |
| · apply hne.2 |
| calc |
| x = D.vertex j := hj |
| _ = D.vertex ((graph.dst e)) := congrArg D.vertex he |
| _ = P.vertices (i+1) := by simpa only [hii] using P.last.symm |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem exists_vertex_piece_clearance [Fintype Vertex] [Fintype Edge] (D : Paper170.ContinuousDrawing graph) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ e, ∀ Q : PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e))), (P e).pieces ≤ Q.pieces) : |
| ∃ r : ℝ, 0 < r ∧ |
| (∀ j k, j ≠ k → Disjoint (closedRoundDisk (D.vertex j) r) (closedRoundDisk (D.vertex k) r)) ∧ |
| (∀ j e (i : Fin (P e).pieces), |
| ¬ ((j=(graph.src e) ∧ i.val=0) ∨ (j=(graph.dst e) ∧ i.val+1=(P e).pieces)) → |
| Disjoint (closedRoundDisk (D.vertex j) r) |
| (segment ℝ ((P e).vertices i.val) ((P e).vertices (i.val+1)))) := by |
| classical |
| let Bad := Σ j : Vertex, Σ e : Edge, |
| {i : Fin (P e).pieces // ¬ ((j=(graph.src e) ∧ i.val=0) ∨ |
| (j=(graph.dst e) ∧ i.val+1=(P e).pieces))} |
| let K : Bad → Set Plane := fun t => |
| segment ℝ ((P t.2.1).vertices t.2.2.val.val) ((P t.2.1).vertices (t.2.2.val.val+1)) |
| have hK (t : Bad) : IsCompact (K t) := by |
| dsimp [K] |
| rw [← Path.range_segment] |
| exact isCompact_range (Path.segment _ _).continuous |
| have hnot (t : Bad) : D.vertex t.1 ∉ K t := by |
| intro hx |
| exact t.2.2.property (drawing_chain_piece_vertex D t.2.1 (P t.2.1) |
| (hmin t.2.1) t.2.2.val.val t.2.2.val.isLt t.1 hx) |
| obtain ⟨ρ,hρ,hclear⟩ := exists_round_disk_clearance (fun t : Bad => D.vertex t.1) K hK hnot |
| obtain ⟨σ,hσ,hdis⟩ := exists_disjoint_round_disks D.vertex D.vertex_injective |
| refine ⟨min ρ σ,lt_min hρ hσ,?_,?_⟩ |
| · intro j k hjk |
| exact (hdis j k hjk).mono |
| (closedRoundDisk_mono _ (le_of_lt (lt_min hρ hσ)) (min_le_right _ _)) |
| (closedRoundDisk_mono _ (le_of_lt (lt_min hρ hσ)) (min_le_right _ _)) |
| · intro j e i hi |
| exact (hclear ⟨j,e,⟨i,hi⟩⟩).mono_left |
| (closedRoundDisk_mono _ (le_of_lt (lt_min hρ hσ)) (min_le_left _ _)) |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem drawing_chain_startPort_interior (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ Q : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) |
| (D.vertex ((graph.dst e))), P.pieces ≤ Q.pieces) |
| (r : ℝ) (hr : 0 < r) (hshort : r^2 < squaredLength (P.vertices 1-P.vertices 0)) : |
| P.startPort r ∈ Paper170.edgeInterior D e := by |
| exact drawing_chain_piece_open_subset_interior D e P hmin 0 (drawing_chain_pieces_pos D e P) |
| (radialPort_mem_openSegment _ _ r hr hshort) |
|
|
| theorem drawing_chain_endPort_interior (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ Q : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) |
| (D.vertex ((graph.dst e))), P.pieces ≤ Q.pieces) |
| (r : ℝ) (hr : 0 < r) |
| (hshort : r^2 < squaredLength (P.vertices P.pieces-P.vertices (P.pieces-1))) : |
| P.endPort r ∈ Paper170.edgeInterior D e := by |
| have hp := drawing_chain_pieces_pos D e P |
| have hs : r^2 < squaredLength (P.vertices (P.pieces-1)-P.vertices P.pieces) := by |
| rwa [squaredLength_sub_comm] |
| have hx := openSegmentTrace_reverse_mem _ _ _ (radialPort_mem_openSegment _ _ r hr hs) |
| apply drawing_chain_piece_open_subset_interior D e P hmin (P.pieces-1) (by omega) |
| simpa only [PolygonalChain.endPort,show P.pieces-1+1=P.pieces by omega] using hx |
|
|
| theorem drawing_chain_startPort_circle (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ Q : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) |
| (D.vertex ((graph.dst e))), P.pieces ≤ Q.pieces) (r : ℝ) : |
| squaredLength (P.startPort r-D.vertex ((graph.src e)))=r^2 := by |
| have hp := drawing_chain_pieces_pos D e P |
| have h := radialPort_on_circle (P.vertices 0) (P.vertices 1) |
| (P.minimum_vertices_ne hmin 0 1 (by omega) hp) r |
| simpa only [PolygonalChain.startPort,P.first] using h |
|
|
| theorem drawing_chain_endPort_circle (D : Paper170.ContinuousDrawing graph) |
| (e : Edge) |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ Q : PolygonalChain (Set.range (D.edge e)) (D.vertex ((graph.src e))) |
| (D.vertex ((graph.dst e))), P.pieces ≤ Q.pieces) (r : ℝ) : |
| squaredLength (P.endPort r-D.vertex ((graph.dst e)))=r^2 := by |
| have hp := drawing_chain_pieces_pos D e P |
| have h := radialPort_on_circle (P.vertices P.pieces) (P.vertices (P.pieces-1)) |
| (Ne.symm (P.minimum_vertices_ne hmin (P.pieces-1) P.pieces (by omega) (by omega))) r |
| simpa only [PolygonalChain.endPort,P.last] using h |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem exists_vertex_trace_disks [Fintype Vertex] [Fintype Edge] (D : Paper170.ContinuousDrawing graph) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ e, ∀ Q : PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e))), (P e).pieces ≤ Q.pieces) : |
| ∃ r : ℝ, 0 < r ∧ |
| (∀ j k, j ≠ k → Disjoint (closedRoundDisk (D.vertex j) r) (closedRoundDisk (D.vertex k) r)) ∧ |
| (∀ j e x, x ∈ closedRoundDisk (D.vertex j) r → |
| (x ∈ Set.range (D.edge e) ↔ |
| (j=(graph.src e) ∧ x ∈ segment ℝ ((P e).vertices 0) ((P e).vertices 1)) ∨ |
| (j=(graph.dst e) ∧ x ∈ segment ℝ ((P e).vertices ((P e).pieces-1)) ((P e).vertices (P e).pieces)))) := by |
| classical |
| obtain ⟨r,hr,hdis,hclear⟩ := exists_vertex_piece_clearance D P hmin |
| refine ⟨r,hr,hdis,?_⟩ |
| intro j e x hxd |
| have hpos := drawing_chain_pieces_pos D e (P e) |
| constructor |
| · intro hx |
| rw [← drawing_chain_range_eq D e (P e)] at hx |
| obtain ⟨i,hi,hxi⟩ := polygonalArcPath_range_piece (P e).vertices (P e).pieces hpos x hx |
| have hinc : (j=(graph.src e) ∧ i=0) ∨ (j=(graph.dst e) ∧ i+1=(P e).pieces) := by |
| by_contra h |
| exact Set.disjoint_left.mp (hclear j e ⟨i,hi⟩ h) hxd hxi |
| rcases hinc with ⟨hj,hii⟩ | ⟨hj,hii⟩ |
| · exact Or.inl ⟨hj,by simpa only [hii] using hxi⟩ |
| · have hlast : i=(P e).pieces-1 := by omega |
| exact Or.inr ⟨hj,by simpa only [hlast,show (P e).pieces-1+1=(P e).pieces by omega] using hxi⟩ |
| · rintro (⟨hj,hx⟩ | ⟨hj,hx⟩) |
| · exact (P e).inside 0 hpos hx |
| · have hi : (P e).pieces-1 < (P e).pieces := by omega |
| apply (P e).inside ((P e).pieces-1) hi |
| simpa only [show (P e).pieces-1+1=(P e).pieces by omega] using hx |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| def IncidentEdges (graph : Paper170.Graph Vertex Edge) (j : Vertex) := |
| {e : Edge // j=(graph.src e) ∨ j=(graph.dst e)} |
|
|
| instance incidentEdgesFintype [Fintype Edge] (j : Vertex) : Fintype (IncidentEdges graph j) := by |
| classical |
| unfold IncidentEdges |
| infer_instance |
|
|
| theorem incident_edges_adjacent (j : Vertex) (e f : IncidentEdges graph j) : |
| graph.src e.val=graph.src f.val ∨ graph.src e.val=graph.dst f.val ∨ |
| graph.dst e.val=graph.src f.val ∨ graph.dst e.val=graph.dst f.val := by |
| rcases e.property with he | he <;> rcases f.property with hf | hf |
| · exact Or.inl (he.symm.trans hf) |
| · exact Or.inr (Or.inl (he.symm.trans hf)) |
| · exact Or.inr (Or.inr (Or.inl (he.symm.trans hf))) |
| · exact Or.inr (Or.inr (Or.inr (he.symm.trans hf))) |
| |
| def drawingSpokePort (D : Paper170.ContinuousDrawing graph) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (r : ℝ) (j : Vertex) (e : IncidentEdges graph j) : Plane := |
| if j=graph.src e.val then (P e.val).startPort r else (P e.val).endPort r |
|
|
| theorem drawingSpokePort_interior (D : Paper170.ContinuousDrawing graph) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ e, ∀ Q : PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e))), (P e).pieces ≤ Q.pieces) |
| (r : ℝ) (hr : 0 < r) |
| (hshort : ∀ e, r^2 < squaredLength ((P e).vertices 1-(P e).vertices 0) ∧ |
| r^2 < squaredLength ((P e).vertices (P e).pieces-(P e).vertices ((P e).pieces-1))) |
| (j : Vertex) (e : IncidentEdges graph j) : |
| drawingSpokePort D P r j e ∈ Paper170.edgeInterior D e.val := by |
| unfold drawingSpokePort |
| split_ifs |
| · exact drawing_chain_startPort_interior D e.val (P e.val) (hmin e.val) r hr (hshort e.val).1 |
| · exact drawing_chain_endPort_interior D e.val (P e.val) (hmin e.val) r hr (hshort e.val).2 |
|
|
| theorem drawingSpokePort_circle (D : Paper170.ContinuousDrawing graph) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ e, ∀ Q : PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e))), (P e).pieces ≤ Q.pieces) |
| (r : ℝ) (j : Vertex) (e : IncidentEdges graph j) : |
| squaredLength (drawingSpokePort D P r j e-D.vertex j)=r^2 := by |
| unfold drawingSpokePort |
| split_ifs with h |
| · simpa only [h] using drawing_chain_startPort_circle D e.val (P e.val) (hmin e.val) r |
| · have he := e.property.resolve_left h |
| simpa only [he] using drawing_chain_endPort_circle D e.val (P e.val) (hmin e.val) r |
|
|
| theorem drawingSpokePort_injective (D : Paper170.AdmissibleDrawing graph) (hnorm : NoAdjacentCrossings D) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ e, ∀ Q : PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e))), (P e).pieces ≤ Q.pieces) |
| (r : ℝ) (hr : 0 < r) |
| (hshort : ∀ e, r^2 < squaredLength ((P e).vertices 1-(P e).vertices 0) ∧ |
| r^2 < squaredLength ((P e).vertices (P e).pieces-(P e).vertices ((P e).pieces-1))) |
| (j : Vertex) : Function.Injective (drawingSpokePort D.toContinuousDrawing P r j) := by |
| intro e f heq |
| by_contra hne |
| have hef : e.val ≠ f.val := fun h => hne (Subtype.ext h) |
| have hp := drawingSpokePort_interior D.toContinuousDrawing P hmin r hr hshort j e |
| have hq := drawingSpokePort_interior D.toContinuousDrawing P hmin r hr hshort j f |
| rw [← heq] at hq |
| have hh : drawingSpokePort D.toContinuousDrawing P r j e ∈ |
| Paper170.edgeInterior D.toContinuousDrawing e.val ∩ Paper170.edgeInterior D.toContinuousDrawing f.val := ⟨hp,hq⟩ |
| rw [hnorm.eq_empty e.val f.val hef (incident_edges_adjacent j e f)] at hh |
| exact hh |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem exists_vertex_spoke_bound [Fintype Edge] (D : Paper170.ContinuousDrawing graph) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ e, ∀ Q : PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e))), (P e).pieces ≤ Q.pieces) |
| (ρ : ℝ) (hρ : 0 < ρ) : |
| ∃ r : ℝ, 0 < r ∧ r ≤ ρ ∧ ∀ e, |
| r^2 < squaredLength ((P e).vertices 1-(P e).vertices 0) ∧ |
| r^2 < squaredLength ((P e).vertices (P e).pieces-(P e).vertices ((P e).pieces-1)) := by |
| let L : Edge → ℝ := fun e => |
| min (squaredLength ((P e).vertices 1-(P e).vertices 0)) |
| (squaredLength ((P e).vertices (P e).pieces-(P e).vertices ((P e).pieces-1))) |
| have hL (e : Edge) : 0 < L e := by |
| have hp := drawing_chain_pieces_pos D e (P e) |
| have hlast := (P e).minimum_piece_ne_zero (hmin e) ((P e).pieces-1) (by omega) |
| rw [show (P e).pieces-1+1=(P e).pieces by omega] at hlast |
| exact lt_min (squaredLength_pos _ ((P e).minimum_piece_ne_zero (hmin e) 0 hp)) |
| (squaredLength_pos _ hlast) |
| obtain ⟨r,hr,hrρ,hbound⟩ := exists_small_squared_radius ρ hρ L hL |
| exact ⟨r,hr,hrρ,fun e => ⟨(hbound e).trans_le (min_le_left _ _), |
| (hbound e).trans_le (min_le_right _ _)⟩⟩ |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| theorem exists_short_vertex_radius [Fintype Edge] (D : Paper170.ContinuousDrawing graph) |
| (P : ∀ e : Edge, PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (hmin : ∀ e, ∀ Q : PolygonalChain (Set.range (D.edge e)) |
| (D.vertex ((graph.src e))) (D.vertex ((graph.dst e))), (P e).pieces ≤ Q.pieces) |
| (ρ : ℝ) (hρ : 0 < ρ) : |
| ∃ r : ℝ, 0 < r ∧ r ≤ ρ ∧ ∀ e, |
| 4*r^2 < squaredLength ((P e).vertices 1-(P e).vertices 0) ∧ |
| 4*r^2 < squaredLength ((P e).vertices (P e).pieces-(P e).vertices ((P e).pieces-1)) := by |
| obtain ⟨s,hs,hsρ,hbound⟩ := exists_vertex_spoke_bound D P hmin ρ hρ |
| refine ⟨s/2,by positivity,by linarith,?_⟩ |
| intro e |
| obtain ⟨hfirst,hlast⟩ := hbound e |
| constructor <;> nlinarith |
| |
| end Paper170Geometry |
| |
| end |
| end |
| |
| end OAI |
| |
