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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 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.Combinatorics.Crossing.BalancedAxisDrawing
import OAI.Combinatorics.Crossing.BalancedRayPairs
namespace OAI
noncomputable section
namespace Zarankiewicz
theorem positive_scaled_eq_sign (a b t u : ℝ) (ht : 0 < t) (hu : 0 < u) (he : t*a=u*b) :
(0 < a ↔ 0 < b) := by
have hh : (0 < t*a ↔ 0 < a) := mul_pos_iff_of_pos_left ht
have hh' : (0 < u*b ↔ 0 < b) := mul_pos_iff_of_pos_left hu
rw [he] at hh
exact hh.symm.trans hh'
theorem axis_meeting_abs_order (a b c d t u : ℝ)
(ha : 0 < |a|) (hd : 0 < |d|) (hac : |a| < |c|)
(ht : t ∈ Set.Ioo (0 : ℝ) 1) (hu : u ∈ Set.Ioo (0 : ℝ) 1)
(he : axisSegmentPoint a b t=axisSegmentPoint c d u) : |d| < |b| := by
have hx := congrArg (fun z : Plane => |z.1|) he
have hy := congrArg (fun z : Plane => |z.2|) he
simp only [axisSegmentPoint_coordinates,abs_mul,abs_of_pos ht.1,abs_of_pos hu.1,
abs_of_pos (sub_pos.mpr ht.2),abs_of_pos (sub_pos.mpr hu.2)] at hx hy
have htu : t < u := by
by_contra hnot
have hgap := mul_pos (sub_pos.mpr hac) (sub_pos.mpr hu.2)
have hdiff := mul_nonneg (sub_nonneg.mpr (le_of_not_gt hnot)) ha.le
nlinarith
have hgap := mul_pos (sub_pos.mpr htu) hd
have htb : t*(|b|-|d|) > 0 := by nlinarith [hx]
exact sub_pos.mp ((mul_pos_iff_of_pos_left ht.1).mp htb)
namespace BalancedAxisRealization
variable {m n : ℕ} (R : BalancedAxisRealization m n)
theorem meeting_same_rays (i k : Fin m) (p q : Fin n) (x : Plane)
(hx : x ∈ edgeInterior R.drawing.toContinuousDrawing (i,p))
(hy : x ∈ edgeInterior R.drawing.toContinuousDrawing (k,q)) :
(i.val < m/2 ↔ k.val < m/2) ∧ (p.val < n/2 ↔ q.val < n/2) := by
rw [R.drawing_edgeInterior] at hx hy
obtain ⟨t,ht,he⟩ := hx
obtain ⟨u,hu,hf⟩ := hy
have hmeet : axisSegmentPoint (R.vertical i) (R.horizontal p) t=
axisSegmentPoint (R.vertical k) (R.horizontal q) u := he.trans hf.symm
have hcoordx := congrArg Prod.fst hmeet
have hcoordy := congrArg Prod.snd hmeet
simp only [axisSegmentPoint_coordinates] at hcoordx hcoordy
have hposx := positive_scaled_eq_sign _ _ _ _ ht.1 hu.1 hcoordx
have hposy := positive_scaled_eq_sign _ _ _ _ (sub_pos.mpr ht.2) (sub_pos.mpr hu.2) hcoordy
exact ⟨(axisInterval_sign m i _ (R.vertical_interval i)).1.symm.trans
(hposy.trans (axisInterval_sign m k _ (R.vertical_interval k)).1),
(axisInterval_sign n p _ (R.horizontal_interval p)).1.symm.trans
(hposx.trans (axisInterval_sign n q _ (R.horizontal_interval q)).1)⟩
theorem meeting_reverse_horizontal_order (i k : Fin m) (p q : Fin n) (hik : i < k) (x : Plane)
(hx : x ∈ edgeInterior R.drawing.toContinuousDrawing (i,p))
(hy : x ∈ edgeInterior R.drawing.toContinuousDrawing (k,q)) : q < p := by
rw [R.drawing_edgeInterior] at hx hy
obtain ⟨t,ht,he⟩ := hx
obtain ⟨u,hu,hf⟩ := hy
have hmeet : axisSegmentPoint (R.vertical i) (R.horizontal p) t=
axisSegmentPoint (R.vertical k) (R.horizontal q) u := he.trans hf.symm
have hh := axis_meeting_abs_order _ _ _ _ t u (abs_pos.mpr (R.vertical_ne_zero i))
(abs_pos.mpr (R.horizontal_ne_zero q)) (R.vertical_abs_strictMono hik) ht hu hmeet
by_contra hnot
have hle := R.horizontal_abs_strictMono.monotone (le_of_not_gt hnot)
exact not_lt_of_ge hle hh
theorem meeting_balanced_pairs (i k : Fin m) (p q : Fin n) (hik : i < k) (x : Plane)
(hx : x ∈ edgeInterior R.drawing.toContinuousDrawing (i,p))
(hy : x ∈ edgeInterior R.drawing.toContinuousDrawing (k,q)) :
(i,k) ∈ balancedRayPairs m ∧ (q,p) ∈ balancedRayPairs n := by
obtain ⟨hvert,hhor⟩ := R.meeting_same_rays i k p q x hx hy
exact ⟨(mem_balancedRayPairs m i k).mpr ⟨hik,hvert⟩,
(mem_balancedRayPairs n q p).mpr ⟨R.meeting_reverse_horizontal_order i k p q hik x hx hy,hhor.symm⟩⟩
end BalancedAxisRealization
end Zarankiewicz
end
end OAI
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