| import OAI.Combinatorics.Crossing.CircleOffset |
| import OAI.Combinatorics.Crossing.CircleOffsetParameter |
| import OAI.Combinatorics.Crossing.MovingSegmentStability |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz |
|
|
| def circleTangentOffset (u : Plane) (δ : ℝ) : Plane := u+δ • rotateQuarter u |
|
|
| theorem squaredLength_radial (u : Plane) (hu : squaredLength u=1) (a b : ℝ) : |
| squaredLength (a • u+b • rotateQuarter u)=a^2+b^2 := by |
| have he : squaredLength (a • u+b • rotateQuarter u)=(a^2+b^2)*squaredLength u := by |
| dsimp [squaredLength,rotateQuarter] |
| ring |
| rw [he,hu,mul_one] |
|
|
| theorem circleConnector_parameterization (u : Plane) (δ t : ℝ) : |
| segmentPoint (circleOffset u δ) (circleTangentOffset u δ) t = |
| ((1-t)*Real.sqrt (1-δ^2)+t) • u+δ • rotateQuarter u := by |
| apply Prod.ext <;> dsimp [segmentPoint,circleOffset,circleTangentOffset,rotateQuarter] <;> ring |
|
|
| theorem circleConnector_outside (u : Plane) (hu : squaredLength u=1) |
| (δ : ℝ) (hδ : |δ|≤1) (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) 1) : |
| 1 ≤ squaredLength (segmentPoint (circleOffset u δ) (circleTangentOffset u δ) t) := by |
| have hnon : 0 ≤ 1-δ^2 := by have hh := abs_le.mp hδ; nlinarith |
| have hs := Real.sq_sqrt hnon |
| have hs0 := Real.sqrt_nonneg (1-δ^2) |
| have hs1 : Real.sqrt (1-δ^2) ≤ 1 := by nlinarith [sq_nonneg δ] |
| have hA : Real.sqrt (1-δ^2) ≤ (1-t)*Real.sqrt (1-δ^2)+t := by |
| have hmul := mul_nonneg ht.1 (sub_nonneg.mpr hs1) |
| nlinarith |
| rw [circleConnector_parameterization,squaredLength_radial u hu] |
| nlinarith |
|
|
| theorem circleConnector_strictly_outside (u : Plane) (hu : squaredLength u=1) |
| (δ : ℝ) (hδ : |δ|<1) (hne : δ ≠ 0) (t : ℝ) (ht : 0<t) : |
| 1 < squaredLength (segmentPoint (circleOffset u δ) (circleTangentOffset u δ) t) := by |
| have hnon : 0 ≤ 1-δ^2 := by have hh := abs_lt.mp hδ; nlinarith |
| have hs := Real.sq_sqrt hnon |
| have hs0 := Real.sqrt_nonneg (1-δ^2) |
| have hs1 : Real.sqrt (1-δ^2) < 1 := by nlinarith [sq_pos_of_ne_zero hne] |
| have hA : Real.sqrt (1-δ^2) < (1-t)*Real.sqrt (1-δ^2)+t := by |
| have hmul := mul_pos ht (sub_pos.mpr hs1) |
| nlinarith |
| rw [circleConnector_parameterization,squaredLength_radial u hu] |
| nlinarith |
|
|
| theorem circleConnector_side (u : Plane) (hu : squaredLength u=1) (δ t : ℝ) : |
| det2 u (segmentPoint (circleOffset u δ) (circleTangentOffset u δ) t)=δ := by |
| rw [circleConnector_parameterization,det2_add_right,det2_smul_right,det2_self,mul_zero] |
| have he : det2 u (rotateQuarter u)=squaredLength u := by |
| dsimp [det2,rotateQuarter,squaredLength] |
| ring |
| rw [det2_smul_right,he,hu,mul_one,zero_add] |
|
|
| end Zarankiewicz |
|
|
| end |
|
|
| end OAI |
|
|
