File size: 3,299 Bytes
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 71 | import OAI.Combinatorics.Crossing.AxisAdmissibility
import OAI.Combinatorics.Crossing.BalancedAxisIntervals
namespace OAI
noncomputable section
namespace Zarankiewicz
structure BalancedAxisRealization (m n : ℕ) where
vertical : Fin m → ℝ
horizontal : Fin n → ℝ
vertical_interval : ∀ i, vertical i ∈ Set.Ioo (axisIntervalLo m i) (axisIntervalHi m i)
horizontal_interval : ∀ p, horizontal p ∈ Set.Ioo (axisIntervalLo n p) (axisIntervalHi n p)
generic : ∀ e f g : BipartiteEdge m n, AxisIndependentTriple e f g →
axisConcurrency (vertical e.1) (horizontal e.2) (vertical f.1) (horizontal f.2)
(vertical g.1) (horizontal g.2) ≠ 0
theorem nonempty_balancedAxisRealization (m n : ℕ) : Nonempty (BalancedAxisRealization m n) := by
let lo : BipartiteVertex m n → ℝ
| .inl i => axisIntervalLo m i
| .inr p => axisIntervalLo n p
let hi : BipartiteVertex m n → ℝ
| .inl i => axisIntervalHi m i
| .inr p => axisIntervalHi n p
have hinterval : ∀ j, lo j < hi j := by
rintro (i|p)
· exact axisInterval_nonempty m i
· exact axisInterval_nonempty n p
obtain ⟨x,hx,hgeneric⟩ := exists_axis_generic_positions lo hi hinterval
exact ⟨⟨fun i => x (Sum.inl i),fun p => x (Sum.inr p),fun i => hx (Sum.inl i),
fun p => hx (Sum.inr p),hgeneric⟩⟩
namespace BalancedAxisRealization
variable {m n : ℕ} (R : BalancedAxisRealization m n)
theorem vertical_injective : Function.Injective R.vertical := axisInterval_injective m _ R.vertical_interval
theorem horizontal_injective : Function.Injective R.horizontal := axisInterval_injective n _ R.horizontal_interval
theorem vertical_ne_zero (i : Fin m) : R.vertical i ≠ 0 := axisInterval_ne_zero m i _ (R.vertical_interval i)
theorem horizontal_ne_zero (p : Fin n) : R.horizontal p ≠ 0 := axisInterval_ne_zero n p _ (R.horizontal_interval p)
theorem vertical_abs_strictMono : StrictMono (fun i => |R.vertical i|) := axisInterval_abs_strictMono m _ R.vertical_interval
theorem horizontal_abs_strictMono : StrictMono (fun p => |R.horizontal p|) := axisInterval_abs_strictMono n _ R.horizontal_interval
def drawing : AdmissibleDrawing m n :=
axisAdmissibleDrawing R.vertical R.horizontal R.vertical_injective R.horizontal_injective
R.vertical_ne_zero R.horizontal_ne_zero R.generic
theorem drawing_noAdjacent : NoAdjacentCrossings R.drawing :=
axisAdmissibleDrawing_noAdjacent _ _ _ _ _ _ _
theorem drawing_vertex_left (i : Fin m) : R.drawing.vertex (Sum.inl i)=(0,R.vertical i) := rfl
theorem drawing_vertex_right (p : Fin n) : R.drawing.vertex (Sum.inr p)=(R.horizontal p,0) := rfl
theorem drawing_edge (e : BipartiteEdge m n) :
R.drawing.edge e=Path.segment (0,R.vertical e.1) (R.horizontal e.2,0) := rfl
theorem drawing_edgeInterior (e : BipartiteEdge m n) :
edgeInterior R.drawing.toContinuousDrawing e=openSegmentTrace (0,R.vertical e.1) (R.horizontal e.2,0) :=
axisContinuousDrawing_interior R.vertical R.horizontal R.vertical_injective R.horizontal_injective
R.vertical_ne_zero R.horizontal_ne_zero e
end BalancedAxisRealization
theorem exists_axis_admissible (m n : ℕ) : Nonempty (AdmissibleDrawing m n) := by
obtain ⟨R⟩ := nonempty_balancedAxisRealization m n
exact ⟨R.drawing⟩
end Zarankiewicz
end
end OAI
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