| import OAI.Combinatorics.Crossing.ParallelDirection |
| import Mathlib.Analysis.Convex.Basic |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace Zarankiewicz |
|
|
| theorem collinear_segment_subset_union (a b c : Plane) (hac : a ≠ c) |
| (h : det2 (b-a) (c-b)=0) : |
| segment ℝ a c ⊆ segment ℝ a b ∪ segment ℝ b c := by |
| have hd : det2 (c-a) (b-a)=0 := by |
| have he : det2 (c-a) (b-a) = -det2 (b-a) (c-b) := by dsimp [det2]; ring |
| rw [he,h,neg_zero] |
| have hp := vector_parallel_of_det_zero (c-a) (b-a) (sub_ne_zero.mpr (Ne.symm hac)) hd |
| have hb : b ∈ Set.range (AffineMap.lineMap a c : ℝ → Plane) := by |
| refine ⟨dot2 (c-a) (b-a)/squaredLength (c-a),?_⟩ |
| rw [AffineMap.lineMap_apply] |
| change (dot2 (c-a) (b-a)/squaredLength (c-a)) • (c-a)+a=b |
| rw [← hp,sub_add_cancel] |
| intro x hx |
| rw [← insert_endpoints_openSegment] at hx |
| rcases hx with hxa | hxc | hxo |
| · subst x |
| exact Or.inl (left_mem_segment ℝ a b) |
| · subst x |
| exact Or.inr (right_mem_segment ℝ b c) |
| · rcases openSegment_subset_union a c hb hxo with hxb | hx' |
| · subst x |
| exact Or.inl (right_mem_segment ℝ a b) |
| · exact hx'.elim (fun hh => Or.inl (openSegment_subset_segment ℝ a b hh)) |
| (fun hh => Or.inr (openSegment_subset_segment ℝ b c hh)) |
|
|
| end Zarankiewicz |
|
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| end |
|
|
| end OAI |
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|
