| import OAI.Combinatorics.Crossing.MinimumChainVertexPieces |
| import OAI.Combinatorics.Crossing.OpenSegmentEndpoints |
| import OAI.Combinatorics.Crossing.DrawingRangeCrossings |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz |
|
|
| theorem drawing_chain_piece_open_subset_interior {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))) |
| (hmin : ∀ Q : PolygonalChain (Set.range (D.edge e)) (D.vertex (.inl e.1)) |
| (D.vertex (.inr e.2)), P.pieces ≤ Q.pieces) |
| (i : ℕ) (hi : i < P.pieces) : |
| openSegmentTrace (P.vertices i) (P.vertices (i+1)) ⊆ 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 (.inl e.1) := 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 (.inr e.2) := congrArg D.vertex he |
| _ = P.vertices (i+1) := by simpa only [hii] using P.last.symm |
|
|
| end Zarankiewicz |
|
|
| end |
|
|
| end OAI |
|
|
