| import OAI.Combinatorics.Crossing.AxisCrossingOrder |
| import OAI.Combinatorics.Crossing.EvenCrossingLowerBound |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz.BalancedAxisRealization |
| open scoped BigOperators Finset |
| variable {m n : ℕ} (R : BalancedAxisRealization m n) |
|
|
| def crossingCandidates (_R : BalancedAxisRealization m n) : Finset ((Fin m × Fin m) × (Fin n × Fin n)) := |
| balancedRayPairs m ×ˢ balancedRayPairs n |
|
|
| def candidateCrossings (v : (Fin m × Fin m) × (Fin n × Fin n)) : Finset Plane := |
| pairCrossings R.drawing (v.1.1,v.2.2) (v.1.2,v.2.1) |
|
|
| theorem pairCrossings_card_le_one (e f : BipartiteEdge m n) : (pairCrossings R.drawing e f).card ≤ 1 := by |
| classical |
| by_cases hef : e=f |
| · subst f |
| simp [pairCrossings_self] |
| · apply Finset.card_le_one.mpr |
| intro x hx y hy |
| have hsub := axisContinuousDrawing_pair_subsingleton R.vertical R.horizontal R.vertical_injective |
| R.horizontal_injective R.vertical_ne_zero R.horizontal_ne_zero e f hef |
| exact hsub (Finset.mem_filter.mp hx).2.2 (Finset.mem_filter.mp hy).2.2 |
|
|
| theorem crossings_covered_by_candidates : |
| R.drawing.crossings_finite.toFinset ⊆ R.crossingCandidates.biUnion R.candidateCrossings := by |
| classical |
| intro x hx |
| have hcross : x ∈ crossingPoints R.drawing.toContinuousDrawing := by simpa using hx |
| obtain ⟨e,f,hef,he,hf⟩ := hcross |
| have hind := axisContinuousDrawing_meeting_independent R.vertical R.horizontal R.vertical_injective |
| R.horizontal_injective R.vertical_ne_zero R.horizontal_ne_zero e f hef x he hf |
| rcases lt_or_gt_of_ne hind.1 with hlt | hgt |
| · have hc := R.meeting_balanced_pairs e.1 f.1 e.2 f.2 hlt x he hf |
| apply Finset.mem_biUnion.mpr |
| refine ⟨((e.1,f.1),(f.2,e.2)),Finset.mem_product.mpr hc,?_⟩ |
| apply Finset.mem_filter.mpr |
| exact ⟨hx,hef,he,hf⟩ |
| · have hc := R.meeting_balanced_pairs f.1 e.1 f.2 e.2 hgt x hf he |
| apply Finset.mem_biUnion.mpr |
| refine ⟨((f.1,e.1),(e.2,f.2)),Finset.mem_product.mpr hc,?_⟩ |
| apply Finset.mem_filter.mpr |
| exact ⟨hx,Ne.symm hef,hf,he⟩ |
|
|
| theorem crossingCount_le : crossingCount R.drawing ≤ axisPairs m*axisPairs n := by |
| classical |
| calc |
| crossingCount R.drawing ≤ (R.crossingCandidates.biUnion R.candidateCrossings).card := |
| Finset.card_le_card R.crossings_covered_by_candidates |
| _ ≤ ∑ v ∈ R.crossingCandidates, (R.candidateCrossings v).card := Finset.card_biUnion_le |
| _ ≤ ∑ _v ∈ R.crossingCandidates, 1 := by |
| apply Finset.sum_le_sum |
| intro v hv |
| exact R.pairCrossings_card_le_one _ _ |
| _ = R.crossingCandidates.card := by simp |
| _ = axisPairs m*axisPairs n := by |
| rw [crossingCandidates,Finset.card_product,balancedRayPairs_card,balancedRayPairs_card] |
|
|
| theorem crossingCount_eq : crossingCount R.drawing=axisPairs m*axisPairs n := |
| Nat.le_antisymm R.crossingCount_le (crossing_lower_bound m n R.drawing) |
|
|
| end Zarankiewicz.BalancedAxisRealization |
|
|
| end |
|
|
| end OAI |
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|
