File size: 3,894 Bytes
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 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import OAI.Combinatorics.Crossing.AdjacentSurgeryRoutes
namespace OAI
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}
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)
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
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 _ _ _ _ _
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)) _ _
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
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 _ _
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 _ _
end Zarankiewicz
end
end OAI
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