| import OAI.Combinatorics.Crossing.OrderedAffineSubdivision |
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| namespace OAI |
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| noncomputable section |
| namespace Zarankiewicz |
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| 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)) |
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| 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)⟩ |
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| end Zarankiewicz |
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| end |
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| end OAI |
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