| 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 | |