| import OAI.Combinatorics.Crossing.CircleParameterChart | |
| import OAI.Combinatorics.Crossing.OffsetVectors | |
| import Mathlib.Analysis.Real.Sqrt | |
| namespace OAI | |
| noncomputable section | |
| namespace Zarankiewicz | |
| def circleOffset (u : Plane) (δ : ℝ) : Plane := Real.sqrt (1-δ^2) • u+δ • rotateQuarter u | |
| theorem circleOffset_on_circle (u : Plane) (hu : squaredLength u=1) (δ : ℝ) (hδ : |δ|≤1) : | |
| squaredLength (circleOffset u δ)=1 := by | |
| have hnon : 0 ≤ 1-δ^2 := by | |
| have h := abs_le.mp hδ | |
| nlinarith | |
| have hs := Real.sq_sqrt hnon | |
| have he : squaredLength (circleOffset u δ)= | |
| ((Real.sqrt (1-δ^2))^2+δ^2)*squaredLength u := by | |
| dsimp [squaredLength,circleOffset,rotateQuarter] | |
| ring | |
| rw [he,hu,hs] | |
| ring | |
| theorem circleOffset_side (u : Plane) (hu : squaredLength u=1) (δ : ℝ) : | |
| det2 u (circleOffset u δ)=δ := by | |
| dsimp [circleOffset,det2,rotateQuarter,squaredLength] at * | |
| linear_combination δ*hu | |
| @[simp] theorem circleOffset_zero (u : Plane) : circleOffset u 0=u := by | |
| simp [circleOffset] | |
| end Zarankiewicz | |
| end | |
| end OAI | |