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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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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