| import OAI.Combinatorics.Crossing.SegmentSides |
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| namespace OAI |
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| noncomputable section |
| namespace Zarankiewicz |
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| def circlePoint (t : ℝ) : Plane := ((1-t^2)/(1+t^2), 2*t/(1+t^2)) |
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| theorem circle_denominator_pos (t : ℝ) : 0 < 1+t^2 := by nlinarith [sq_nonneg t] |
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| theorem circlePoint_on_unit_circle (t : ℝ) : (circlePoint t).1^2+(circlePoint t).2^2 = 1 := by |
| have h := ne_of_gt (circle_denominator_pos t) |
| dsimp [circlePoint] |
| field_simp |
| ring |
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| theorem circlePoint_parameter (t : ℝ) : (circlePoint t).2/(1+(circlePoint t).1) = t := by |
| have h := ne_of_gt (circle_denominator_pos t) |
| have hden : 1+(circlePoint t).1 = 2/(1+t^2) := by |
| dsimp [circlePoint] |
| field_simp |
| ring |
| rw [hden] |
| dsimp [circlePoint] |
| field_simp |
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| theorem circlePoint_injective : Function.Injective circlePoint := by |
| intro a b hab |
| have hh := congrArg (fun z : Plane => z.2/(1+z.1)) hab |
| simpa only [circlePoint_parameter] using hh |
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| theorem circle_orient_formula (a b c : ℝ) : |
| orient2 (circlePoint a) (circlePoint b) (circlePoint c) = |
| 4*(b-a)*(c-a)*(c-b)/((1+a^2)*(1+b^2)*(1+c^2)) := by |
| have ha := ne_of_gt (circle_denominator_pos a) |
| have hb := ne_of_gt (circle_denominator_pos b) |
| have hc := ne_of_gt (circle_denominator_pos c) |
| dsimp [orient2,det2,circlePoint] |
| field_simp |
| ring |
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| theorem circle_orient_pos (a b c : ℝ) (hab : a<b) (hbc : b<c) : |
| 0 < orient2 (circlePoint a) (circlePoint b) (circlePoint c) := by |
| rw [circle_orient_formula] |
| exact div_pos |
| (mul_pos (mul_pos (mul_pos (by norm_num) (sub_pos.mpr hab)) |
| (sub_pos.mpr (hab.trans hbc))) (sub_pos.mpr hbc)) |
| (mul_pos (mul_pos (circle_denominator_pos a) (circle_denominator_pos b)) (circle_denominator_pos c)) |
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| theorem orient2_cycle (a b c : Plane) : orient2 b c a = orient2 a b c := by |
| dsimp [orient2,det2] |
| ring |
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| theorem orient2_swap_ends (a b c : Plane) : orient2 a c b = -orient2 a b c := by |
| exact det2_swap (b-a) (c-a) |
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| theorem circle_chords_alternating_positive (a c b d : ℝ) (hac : a<c) (hcb : c<b) (hbd : b<d) : |
| segmentCrossingSign (circlePoint a) (circlePoint b) (circlePoint c) (circlePoint d) = 1 := by |
| apply segmentCrossingSign_of_positive_sides |
| · rw [orient2_swap_ends] |
| exact neg_neg_of_pos (circle_orient_pos a c b hac hcb) |
| · exact circle_orient_pos a b d (hac.trans hcb) hbd |
| · rw [orient2_cycle] |
| exact circle_orient_pos a c d hac (hcb.trans hbd) |
| · rw [orient2_swap_ends] |
| exact neg_neg_of_pos (circle_orient_pos c b d hcb hbd) |
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| end Zarankiewicz |
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| end |
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| end OAI |
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