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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 | 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
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