| import OAI.Combinatorics.Crossing.ReducedPolygonalPresentation | |
| import OAI.Combinatorics.Crossing.GraphWalkTrace | |
| namespace OAI | |
| noncomputable section | |
| namespace Zarankiewicz | |
| theorem PolygonalChain.piece_subset_path_range {O : Set Plane} {a b : Plane} | |
| (P : PolygonalChain O a b) (i : ℕ) (hi : i < P.pieces) : | |
| segment ℝ (P.vertices i) (P.vertices (i+1)) ⊆ Set.range P.path := by | |
| exact polygonalArcPath_piece_subset_range P.vertices P.pieces i hi | |
| theorem drawing_chain_range_eq {m n : ℕ} (D : ContinuousDrawing m n) | |
| (e : BipartiteEdge m n) | |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex (.inl e.1)) (D.vertex (.inr e.2))) : | |
| 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 {m n : ℕ} (D : ContinuousDrawing m n) | |
| (e : BipartiteEdge m n) | |
| (P : PolygonalChain (Set.range (D.edge e)) (D.vertex (.inl e.1)) (D.vertex (.inr e.2))) : | |
| 0 < P.pieces := by | |
| apply P.pieces_pos_of_ne | |
| intro h | |
| cases D.vertex_injective h | |
| end Zarankiewicz | |
| end | |
| end OAI | |