| import OAI.Combinatorics.Crossing.AdjacentSurgeryDrawing |
| import OAI.Combinatorics.Crossing.PolygonalParametrizationCover |
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| namespace OAI |
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| noncomputable section |
| namespace Zarankiewicz |
| variable {m n : ℕ} {D D' : AdmissibleDrawing m n} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
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| theorem adjacentSurgeryDrawing_hasFiniteAffineCover |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hpq : p ≠ q) (hadj : p.1.1=q.1.1 ∨ p.1.2=q.1.2) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (e : BipartiteEdge m n) : HasFiniteAffineCover ((adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge e) := by |
| classical |
| by_cases hep : e=p.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_first] |
| exact adjacentSurgeryRouteP_hasFiniteAffineCover Kp Kq hm hadj |
| by_cases heq : e=q.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_second Kp Kq hm hpq] |
| exact adjacentSurgeryRouteQ_hasFiniteAffineCover Kp Kq hm hadj |
| · rw [adjacentSurgeryDrawing_other Kp Kq hm hadj hv hemb e hep heq] |
| exact (P e).hasFiniteAffineCover |
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| theorem adjacentSurgeryRoutes_avoid_crossing |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hpq : p ≠ q) (hadj : p.1.1=q.1.1 ∨ p.1.2=q.1.2) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| x ∉ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj) ∧ |
| x ∉ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj) := by |
| have ha := reconnected_paths_avoid_crossing Kp Kq hm hpq x hxp hxq |
| constructor |
| · unfold adjacentSurgeryRouteP |
| split |
| · exact ha.2 |
| · exact ha.1 |
| · unfold adjacentSurgeryRouteQ |
| split |
| · exact ha.1 |
| · exact ha.2 |
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| theorem adjacentSurgeryDrawing_avoids_crossing |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hpq : p ≠ q) (hadj : p.1.1=q.1.1 ∨ p.1.2=q.1.2) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) |
| (hother : ∀ e, e ≠ p.1 → e ≠ q.1 → x ∉ Set.range (D'.edge e)) : |
| ∀ e, x ∉ Set.range ((adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge e) := by |
| have ha := adjacentSurgeryRoutes_avoid_crossing Kp Kq hm hpq hadj x hxp hxq |
| intro e |
| classical |
| by_cases hep : e=p.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_first] |
| exact ha.1 |
| by_cases heq : e=q.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_second Kp Kq hm hpq] |
| exact ha.2 |
| · rw [adjacentSurgeryDrawing_other Kp Kq hm hadj hv hemb e hep heq] |
| exact hother e hep heq |
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| end Zarankiewicz |
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| end |
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| end OAI |
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