| import OAI.Combinatorics.Crossing.CircleChords | |
| namespace OAI | |
| namespace Zarankiewicz | |
| theorem circlePoint_one_add_x (t : ℝ) : 1+(circlePoint t).1 = 2/(1+t^2) := by | |
| have h := ne_of_gt (circle_denominator_pos t) | |
| dsimp [circlePoint] | |
| field_simp | |
| ring | |
| theorem circlePoint_avoids_cut (t : ℝ) : (circlePoint t).1 ≠ -1 := by | |
| have hh : 0 < 1+(circlePoint t).1 := by | |
| rw [circlePoint_one_add_x] | |
| exact div_pos (by norm_num) (circle_denominator_pos t) | |
| linarith | |
| theorem circlePoint_surjective_away_cut (z : Plane) (hz : z.1^2+z.2^2 = 1) (hcut : z.1 ≠ -1) : | |
| circlePoint (z.2/(1+z.1)) = z := by | |
| have hx : 0 < 1+z.1 := by | |
| have hy := sq_nonneg z.2 | |
| have hnonneg : 0 ≤ 1+z.1 := by nlinarith | |
| rcases hnonneg.eq_or_lt with he | he | |
| · exfalso | |
| apply hcut | |
| linarith | |
| · exact he | |
| have hxn := ne_of_gt hx | |
| have hden : 1+(z.2/(1+z.1))^2 = 2/(1+z.1) := by | |
| field_simp | |
| nlinarith | |
| apply Prod.ext | |
| · change (1-(z.2/(1+z.1))^2)/(1+(z.2/(1+z.1))^2) = z.1 | |
| rw [hden] | |
| field_simp | |
| nlinarith [congrArg (fun t : ℝ => (1+z.1)*t) hz] | |
| · change (2*(z.2/(1+z.1)))/(1+(z.2/(1+z.1))^2) = z.2 | |
| rw [hden] | |
| field_simp | |
| end Zarankiewicz | |
| end OAI | |