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