| 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 | |