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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 | import OAI.Combinatorics.Crossing.CircleChords
import OAI.Combinatorics.Crossing.PlanarDeterminant
namespace OAI
namespace Zarankiewicz
theorem circle_det_formula (a b : ℝ) :
det2 (circlePoint a) (circlePoint b) =
2*(b-a)*(1+a*b)/((1+a^2)*(1+b^2)) := by
have ha := ne_of_gt (circle_denominator_pos a)
have hb := ne_of_gt (circle_denominator_pos b)
dsimp [det2,circlePoint]
field_simp
ring
theorem circle_det_pos_iff (a b : ℝ) (h : 0<1+a*b) :
0<det2 (circlePoint a) (circlePoint b) ↔ a<b := by
rw [circle_det_formula,div_pos_iff_of_pos_right (mul_pos (circle_denominator_pos a) (circle_denominator_pos b)),
mul_pos_iff_of_pos_right h,mul_pos_iff_of_pos_left (show (0 : ℝ)<2 by norm_num),sub_pos]
theorem circle_det_neg_iff (a b : ℝ) (h : 0<1+a*b) :
det2 (circlePoint a) (circlePoint b)<0 ↔ b<a := by
have hh : 0<1+b*a := by nlinarith
have he := circle_det_pos_iff b a hh
rw [det2_swap,neg_pos] at he
exact he
end Zarankiewicz
end OAI
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