| import OAI.Combinatorics.Crossing.AxisFiniteCrossings |
| import OAI.Combinatorics.Crossing.AxisGenericPositions |
| import OAI.Combinatorics.Crossing.RowCrossingCount |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz |
| variable {m n : ℕ} |
|
|
| theorem axisContinuousDrawing_meeting_independent (a : Fin m → ℝ) (b : Fin n → ℝ) |
| (hai : Function.Injective a) (hbi : Function.Injective b) |
| (ha : ∀ i, a i ≠ 0) (hb : ∀ p, b p ≠ 0) (e f : BipartiteEdge m n) (hef : e ≠ f) |
| (x : Plane) (hx : x ∈ edgeInterior (axisContinuousDrawing a b hai hbi ha hb) e) |
| (hy : x ∈ edgeInterior (axisContinuousDrawing a b hai hbi ha hb) f) : e.1 ≠ f.1 ∧ e.2 ≠ f.2 := by |
| rw [axisContinuousDrawing_interior] at hx hy |
| obtain ⟨t,ht,he⟩ := hx |
| obtain ⟨u,hu,hf⟩ := hy |
| have hmeet : axisSegmentPoint (a e.1) (b e.2) t=axisSegmentPoint (a f.1) (b f.2) u := he.trans hf.symm |
| constructor |
| · intro hfirst |
| rw [hfirst] at hmeet |
| have hh := axisSegmentPoint_same_vertical (a f.1) (b e.2) (b f.2) t u (ha f.1) ht.1 hmeet |
| exact hef (Prod.ext hfirst (hbi hh)) |
| · intro hsecond |
| rw [hsecond] at hmeet |
| have hh := axisSegmentPoint_same_horizontal (a e.1) (a f.1) (b f.2) t u (hb f.2) ht.2 hmeet |
| exact hef (Prod.ext (hai hh) hsecond) |
|
|
| theorem axisContinuousDrawing_line_equation (a : Fin m → ℝ) (b : Fin n → ℝ) |
| (hai : Function.Injective a) (hbi : Function.Injective b) |
| (ha : ∀ i, a i ≠ 0) (hb : ∀ p, b p ≠ 0) (e : BipartiteEdge m n) (x : Plane) |
| (hx : x ∈ edgeInterior (axisContinuousDrawing a b hai hbi ha hb) e) : |
| a e.1*x.1+b e.2*x.2=a e.1*b e.2 := by |
| rw [axisContinuousDrawing_interior] at hx |
| obtain ⟨t,ht,he⟩ := hx |
| rw [← he] |
| exact axisSegmentPoint_line_equation _ _ _ |
|
|
| def axisAdmissibleDrawing (a : Fin m → ℝ) (b : Fin n → ℝ) |
| (hai : Function.Injective a) (hbi : Function.Injective b) |
| (ha : ∀ i, a i ≠ 0) (hb : ∀ p, b p ≠ 0) |
| (hgeneric : ∀ e f g : BipartiteEdge m n, AxisIndependentTriple e f g → |
| axisConcurrency (a e.1) (b e.2) (a f.1) (b f.2) (a g.1) (b g.2) ≠ 0) : AdmissibleDrawing m n where |
| toContinuousDrawing := axisContinuousDrawing a b hai hbi ha hb |
| crossings_finite := axisContinuousDrawing_crossings_finite a b hai hbi ha hb |
| crossings_proper := axisContinuousDrawing_crossings_proper a b hai hbi ha hb |
| no_triples := by |
| intro e f g hef heg hfg |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| rintro x ⟨⟨he,hf⟩,hg⟩ |
| have h₀ := axisContinuousDrawing_meeting_independent a b hai hbi ha hb e f hef x he hf |
| have h₁ := axisContinuousDrawing_meeting_independent a b hai hbi ha hb e g heg x he hg |
| have h₂ := axisContinuousDrawing_meeting_independent a b hai hbi ha hb f g hfg x hf hg |
| exact hgeneric e f g ⟨h₀.1,h₁.1,h₂.1,h₀.2,h₁.2,h₂.2⟩ |
| (axisConcurrency_zero_of_meeting _ _ _ _ _ _ x |
| (axisContinuousDrawing_line_equation a b hai hbi ha hb e x he) |
| (axisContinuousDrawing_line_equation a b hai hbi ha hb f x hf) |
| (axisContinuousDrawing_line_equation a b hai hbi ha hb g x hg)) |
|
|
| theorem axisAdmissibleDrawing_noAdjacent (a : Fin m → ℝ) (b : Fin n → ℝ) |
| (hai : Function.Injective a) (hbi : Function.Injective b) |
| (ha : ∀ i, a i ≠ 0) (hb : ∀ p, b p ≠ 0) |
| (hgeneric : ∀ e f g : BipartiteEdge m n, AxisIndependentTriple e f g → |
| axisConcurrency (a e.1) (b e.2) (a f.1) (b f.2) (a g.1) (b g.2) ≠ 0) : |
| NoAdjacentCrossings (axisAdmissibleDrawing a b hai hbi ha hb hgeneric) := by |
| intro e f hef hadj |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| rintro x ⟨hx,hy⟩ |
| have hh := axisContinuousDrawing_meeting_independent a b hai hbi ha hb e f hef x hx hy |
| exact hadj.elim hh.1 hh.2 |
|
|
| end Zarankiewicz |
|
|
| end |
|
|
| end OAI |
|
|
