| import OAI.Combinatorics.Crossing.TransversePolygonIntersection |
| import OAI.Combinatorics.Crossing.PolygonTransversalityStability |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz |
|
|
| theorem closed_polygon_intersection_zero (v w : ℕ → Plane) (n m : ℕ) |
| (hc : v n=v 0) (hd : w m=w 0) |
| (ht : ∀ i : Fin n, ∀ j : Fin m, |
| SegmentPairTransverse (v i.val) (v (i.val+1)) (w j.val) (w (j.val+1))) : |
| polygonIntersection v w n m = 0 := by |
| by_cases hm : 1<m |
| · have hm0 : 0<m := by omega |
| have hswap (j : Fin m) (i : Fin n) : |
| SegmentPairTransverse (w j.val) (w (j.val+1)) (v i.val) (v (i.val+1)) := |
| segmentPairTransverse_swap _ _ _ _ (ht i j) |
| obtain ⟨ε,hε,hstable⟩ := exists_polygon_transversality_radius w v m n hswap |
| obtain ⟨p,hnear,hinj,_⟩ := exists_generic_point_family |
| (fun i : Fin 0 => (Fin.elim0 i : Plane)) (fun i : Fin 0 => (Fin.elim0 i : Plane)) |
| (fun i : Fin 0 => (Fin.elim0 i : Plane)) (fun i => Fin.elim0 i) |
| m (fun i => w i.val) (fun _ => ε) (fun _ => hε) |
| let w' := periodicPolygon m hm0 p |
| have hclose : w' m=w' 0 := periodicPolygon_closed m hm0 p |
| have hw' (j : Fin m) : w' (j.val+1) ≠ w' j.val := hinj.ne (cyclicIndex_succ_ne m hm j) |
| obtain ⟨htrans,heq⟩ := hstable w' (periodicPolygon_near m hm0 w hd p ε hnear) |
| have hzero := polygon_intersection_zero_transverse v w' n m hc hclose hw' |
| (fun i j => segmentPairTransverse_swap _ _ _ _ (htrans j i)) |
| have hrev := polygonIntersection_swap v w' n m |
| rw [hzero,neg_zero,heq] at hrev |
| rw [polygonIntersection_swap v w n m] at hrev |
| exact neg_eq_zero.mp hrev |
| · have hm' : m=0 ∨ m=1 := by omega |
| rcases hm' with rfl | rfl |
| · simp [polygonIntersection] |
| · have hz : polygonIntersection w v 1 n = 0 := by |
| simp only [polygonIntersection,Finset.sum_range_one] |
| rw [hd] |
| simp only [segmentCrossingSign_self_left,Finset.sum_const_zero] |
| rw [polygonIntersection_swap v w n 1] at hz |
| exact neg_eq_zero.mp hz |
|
|
| end Zarankiewicz |
|
|
| end |
|
|
| end OAI |
|
|
