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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 | 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
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