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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 | import OAI.Combinatorics.Crossing.CyclePairing
import OAI.Combinatorics.Crossing.FourCyclePairing
namespace OAI
noncomputable section
namespace Zarankiewicz.TruncatedSpokeCoordinates
variable {m n : ℕ} {D : AdmissibleDrawing m n} {V : VertexTruncation D.toContinuousDrawing}
(C : TruncatedSpokeCoordinates V)
theorem fourCycle_pairing_zero (hnorm : NoAdjacentCrossings D)
(hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D)
(i k i₀ : Fin m) (p p' q q' : Fin n) :
edgePairing C.cycleIntersectionMatrix (walkEdgeChain (fourCycleWalk i i₀ p p'))
(walkEdgeChain (fourCycleWalk k i₀ q q'))=0 := by
by_cases hi : i=i₀
· subst i
have hz : walkEdgeChain (fourCycleWalk i₀ i₀ p p')=0 := by rw [fourCycleWalk_chain]; abel
rw [hz]
simp [edgePairing]
by_cases hk : k=i₀
· subst k
have hz : walkEdgeChain (fourCycleWalk i₀ i₀ q q')=0 := by rw [fourCycleWalk_chain]; abel
rw [hz]
simp [edgePairing]
by_cases hp : p=p'
· subst p'
have hz : walkEdgeChain (fourCycleWalk i i₀ p p)=0 := by rw [fourCycleWalk_chain]; abel
rw [hz]
simp [edgePairing]
by_cases hq : q=q'
· subst q'
have hz : walkEdgeChain (fourCycleWalk k i₀ q q)=0 := by rw [fourCycleWalk_chain]; abel
rw [hz]
simp [edgePairing]
exact C.lemma3_1 hnorm hstraight htrans _ _
(fourCycleWalk_isCycle i i₀ p p' hi hp) (fourCycleWalk_isCycle k i₀ q q' hk hq)
theorem lemma3_2 (hnorm : NoAdjacentCrossings D)
(hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D)
(i k i₀ : Fin m) :
∃ r c : Fin n → ℝ, ∀ p q,
blockRectangle C.cycleIntersectionMatrix i k i₀ p q=r p+c q := by
cases n with
| zero => exact ⟨0,0,fun p => Fin.elim0 p⟩
| succ n =>
apply rectangle_additive (blockRectangle C.cycleIntersectionMatrix i k i₀) 0 0
intro p p' q q'
rw [← fourCycle_pairing_rectangle]
exact C.fourCycle_pairing_zero hnorm hstraight htrans i k i₀ p p' q q'
end Zarankiewicz.TruncatedSpokeCoordinates
end
end OAI
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