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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 | 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
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