File size: 18,278 Bytes
97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 | 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
|