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d5b7741 | 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 | 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
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