import OAI.Combinatorics.Crossing.FiniteAffineCover
import OAI.Combinatorics.Crossing.PathSubarc

namespace OAI

noncomputable section
namespace Zarankiewicz

theorem intervalSegment_interpolate (a b u v t : unitInterval) :
    intervalSegment (intervalSegment a b u) (intervalSegment a b v) t=
      intervalSegment a b (intervalSegment u v t) := by
  apply Subtype.ext
  dsimp [intervalSegment]
  ring

theorem segmentPoint_interpolate (a b : Plane) (u v t : ℝ) :
    segmentPoint (segmentPoint a b u) (segmentPoint a b v) t=
      segmentPoint a b ((1-t)*u+t*v) := by
  ext <;> dsimp [segmentPoint] <;> ring

theorem path_affine_restrict {a b : Plane} (γ : Path a b) (l r : unitInterval) (hlr : l < r)
    (haff : ∀ t : unitInterval, γ (intervalSegment l r t)=segmentPoint (γ l) (γ r) t.val)
    (u v : unitInterval) (hu : l ≤ u ∧ u ≤ r) (hv : l ≤ v ∧ v ≤ r) (t : unitInterval) :
    γ (intervalSegment u v t)=segmentPoint (γ u) (γ v) t.val := by
  obtain ⟨u',rfl⟩ := intervalSegment_surjective_interval l r hlr u hu
  obtain ⟨v',rfl⟩ := intervalSegment_surjective_interval l r hlr v hv
  rw [intervalSegment_interpolate,haff,haff,haff,segmentPoint_interpolate]
  rfl

end Zarankiewicz

end

end OAI