File size: 3,171 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 | import OAI.Combinatorics.Crossing.OrderedAffineSubdivision
namespace OAI
noncomputable section
namespace Zarankiewicz
theorem HasFiniteAffineCover.refine {a b : Plane} {γ : Path a b}
(hγ : HasFiniteAffineCover γ) (R : Finset unitInterval) :
∃ P : PolygonalParametrization γ, (R : Set unitInterval) ⊆ Set.range P.knot := by
classical
obtain ⟨S,hSlt,hScov,hSaff⟩ := hγ
let T : Finset unitInterval := insert 0 (insert 1 ((S.image Prod.fst ∪ S.image Prod.snd) ∪ R))
obtain ⟨k,hk,t,ht,h0,h1,hR⟩ := exists_sorted_unitInterval_finset T (by simp [T]) (by simp [T])
have hleft (ab : unitInterval × unitInterval) (hab : ab ∈ S) : ab.1 ∈ Set.range t := by
rw [hR]
exact Finset.mem_insert_of_mem (Finset.mem_insert_of_mem
(Finset.mem_union_left _ (Finset.mem_union_left _ (Finset.mem_image.mpr ⟨ab,hab,rfl⟩))))
have hright (ab : unitInterval × unitInterval) (hab : ab ∈ S) : ab.2 ∈ Set.range t := by
rw [hR]
exact Finset.mem_insert_of_mem (Finset.mem_insert_of_mem
(Finset.mem_union_left _ (Finset.mem_union_right _ (Finset.mem_image.mpr ⟨ab,hab,rfl⟩))))
have haff : ∀ i : Fin k, ∀ u : unitInterval,
γ (intervalSegment (t i.castSucc) (t i.succ) u)=segmentPoint (γ (t i.castSucc)) (γ (t i.succ)) u.val := by
intro i u
let mid : unitInterval := ⟨1/2,by norm_num⟩
let z := intervalSegment (t i.castSucc) (t i.succ) mid
have hz : t i.castSucc < z ∧ z < t i.succ :=
intervalSegment_strict_between _ _ mid (ht (by change i.val < i.val+1; omega)) (by
constructor <;> change (_ : ℝ) < _ <;> norm_num [mid])
obtain ⟨ab,hab,ha,hb⟩ := hScov z
have hlo : ab.1 ≤ t i.castSucc := by
by_contra hn
obtain ⟨j,hj⟩ := hleft ab hab
exact strictMono_no_between_successors t ht i j
⟨by rw [hj]; exact lt_of_not_ge hn,by rw [hj]; exact ha.trans_lt hz.2⟩
have hhi : t i.succ ≤ ab.2 := by
by_contra hn
obtain ⟨j,hj⟩ := hright ab hab
exact strictMono_no_between_successors t ht i j
⟨by rw [hj]; exact hz.1.trans_le hb,by rw [hj]; exact lt_of_not_ge hn⟩
have hle : t i.castSucc ≤ t i.succ := ht.monotone (by change i.val ≤ i.val+1; omega)
exact path_affine_restrict γ ab.1 ab.2 (hSlt ab hab) (hSaff ab hab)
(t i.castSucc) (t i.succ) ⟨hlo,hle.trans hhi⟩ ⟨hlo.trans hle,hhi⟩ u
refine ⟨{pieces:=k,knot:=t,increasing:=ht,first:=h0,last:=h1,affine:=haff},?_⟩
intro u hu
change u ∈ Set.range t
rw [hR]
exact Finset.mem_insert_of_mem (Finset.mem_insert_of_mem (Finset.mem_union_right _ hu))
theorem exists_marked_polygonal_subdivision {m n : ℕ} (D : AdmissibleDrawing m n)
(hpoly : ∀ e, HasFiniteAffineCover (D.edge e)) :
∃ P : ∀ e, PolygonalParametrization (D.edge e),
∀ e t, D.edge e t ∈ drawingMarks D.toFiniteIntersectionDrawing → t ∈ Set.range (P e).knot := by
classical
have hex (e : BipartiteEdge m n) := (hpoly e).refine (marked_parameters_finite D.toFiniteIntersectionDrawing e).toFinset
choose P hP using hex
exact ⟨P,fun e t ht => hP e (by simpa using ht)⟩
end Zarankiewicz
end
end OAI
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