| import OAI.Combinatorics.Crossing.AdjacentSurgeryRoutes |
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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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| def adjacentSurgeryDrawing (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark 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) : ContinuousDrawing m n := |
| D'.toContinuousDrawing.replaceTwoEdges p.1 q.1 |
| (adjacentSurgeryRouteP Kp Kq hm hadj) (adjacentSurgeryRouteQ Kp Kq hm hadj) |
| (adjacentSurgeryRouteP_injective Kp Kq hm hadj hemb) |
| (adjacentSurgeryRouteQ_injective Kp Kq hm hadj hemb) |
| (adjacentSurgeryRouteP_avoids_vertices Kp Kq hm hadj hv hemb) |
| (adjacentSurgeryRouteQ_avoids_vertices Kp Kq hm hadj hv hemb) |
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| theorem adjacentSurgeryDrawing_vertex (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark 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) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).vertex=D'.vertex := rfl |
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| theorem adjacentSurgeryDrawing_first (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark 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) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge p.1=adjacentSurgeryRouteP Kp Kq hm hadj := |
| ContinuousDrawing.replaceTwoRoutes_first _ _ _ _ _ |
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| theorem adjacentSurgeryDrawing_second (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) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge q.1=adjacentSurgeryRouteQ Kp Kq hm hadj := |
| ContinuousDrawing.replaceTwoRoutes_second _ _ _ (fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hm)) _ _ |
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| theorem adjacentSurgeryDrawing_other (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark 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) (hep : e ≠ p.1) (heq : e ≠ q.1) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge e=D'.edge e := |
| ContinuousDrawing.replaceTwoRoutes_other _ _ _ _ _ _ hep heq |
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| theorem adjacentSurgeryRouteP_hasFiniteAffineCover |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hadj : p.1.1=q.1.1 ∨ p.1.2=q.1.2) : |
| HasFiniteAffineCover (adjacentSurgeryRouteP Kp Kq hm hadj) := by |
| unfold adjacentSurgeryRouteP |
| split |
| · exact (reconnectedBackwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
| · exact (reconnectedForwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
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| theorem adjacentSurgeryRouteQ_hasFiniteAffineCover |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hadj : p.1.1=q.1.1 ∨ p.1.2=q.1.2) : |
| HasFiniteAffineCover (adjacentSurgeryRouteQ Kp Kq hm hadj) := by |
| unfold adjacentSurgeryRouteQ |
| split |
| · exact (reconnectedForwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
| · exact (reconnectedBackwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
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| end Zarankiewicz |
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| end |
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| end OAI |
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