File size: 3,068 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 72 73 74 | import OAI.Combinatorics.Crossing.ProperSegmentCrossing
namespace OAI
noncomputable section
namespace Zarankiewicz
def axisSegmentPoint (a b t : ℝ) : Plane := segmentPoint (0,a) (b,0) t
theorem axisSegmentPoint_coordinates (a b t : ℝ) : axisSegmentPoint a b t=(t*b,(1-t)*a) := by
ext <;> simp [axisSegmentPoint,segmentPoint]
theorem axisSegmentPoint_injective (a b : ℝ) (hb : b ≠ 0) : Function.Injective (axisSegmentPoint a b) := by
intro t u htu
have hx := congrArg Prod.fst htu
simp only [axisSegmentPoint_coordinates] at hx
exact mul_right_cancel₀ hb hx
theorem axisSegmentPoint_interior_off_axes (a b t : ℝ) (ha : a ≠ 0) (hb : b ≠ 0)
(ht : t ∈ Set.Ioo (0 : ℝ) 1) :
(axisSegmentPoint a b t).1 ≠ 0 ∧ (axisSegmentPoint a b t).2 ≠ 0 := by
rw [axisSegmentPoint_coordinates]
exact ⟨mul_ne_zero (ne_of_gt ht.1) hb,mul_ne_zero (ne_of_gt (sub_pos.mpr ht.2)) ha⟩
theorem axisSegmentPoint_same_vertical (a b d t u : ℝ) (ha : a ≠ 0)
(ht : 0 < t) (h : axisSegmentPoint a b t=axisSegmentPoint a d u) : b=d := by
have hx := congrArg Prod.fst h
have hy := congrArg Prod.snd h
simp only [axisSegmentPoint_coordinates] at hx hy
have htu : t=u := by have hh := mul_right_cancel₀ ha hy; linarith
rw [← htu] at hx
exact mul_left_cancel₀ (ne_of_gt ht) hx
theorem axisSegmentPoint_same_horizontal (a c b t u : ℝ) (hb : b ≠ 0)
(ht : t < 1) (h : axisSegmentPoint a b t=axisSegmentPoint c b u) : a=c := by
have hx := congrArg Prod.fst h
have hy := congrArg Prod.snd h
simp only [axisSegmentPoint_coordinates] at hx hy
have htu : t=u := mul_right_cancel₀ hb hx
rw [← htu] at hy
exact mul_left_cancel₀ (ne_of_gt (sub_pos.mpr ht)) hy
theorem axisSegmentPoint_meeting_det (a b c d t u : ℝ) (ha : a ≠ 0) (hb : b ≠ 0)
(hne : (a,b) ≠ (c,d)) (h : axisSegmentPoint a b t=axisSegmentPoint c d u) :
det2 ((b,0)-(0,a)) ((d,0)-(0,c)) ≠ 0 := by
intro hz
have hx := congrArg Prod.fst h
have hy := congrArg Prod.snd h
simp only [axisSegmentPoint_coordinates] at hx hy
have hz' : a*d-b*c=0 := by
dsimp [det2] at hz
nlinarith [hz]
have hac : (a-c)*b=0 := by linear_combination b*hy+a*hx+u*hz'
have he : a=c := sub_eq_zero.mp ((mul_eq_zero.mp hac).resolve_right hb)
have hbd : a*(d-b)=0 := by rw [← he] at hz'; nlinarith [hz']
have he' : b=d := (sub_eq_zero.mp ((mul_eq_zero.mp hbd).resolve_left ha)).symm
exact hne (Prod.ext he he')
theorem axisSegmentPoint_meeting_proper (a b c d : ℝ) (ha : a ≠ 0) (hb : b ≠ 0)
(hne : (a,b) ≠ (c,d)) (x : Plane)
(hx : x ∈ openSegmentTrace (0,a) (b,0)) (hy : x ∈ openSegmentTrace (0,c) (d,0)) :
ProperCrossing (openSegmentTrace (0,a) (b,0)) (openSegmentTrace (0,c) (d,0)) x := by
obtain ⟨t,ht,he⟩ := hx
obtain ⟨u,hu,hf⟩ := hy
have hmeet : axisSegmentPoint a b t=axisSegmentPoint c d u := he.trans hf.symm
exact properCrossing_openSegments _ _ _ _ _ (axisSegmentPoint_meeting_det a b c d t u ha hb hne hmeet)
⟨t,ht,he⟩ ⟨u,hu,hf⟩
end Zarankiewicz
end
end OAI
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