File size: 3,515 Bytes
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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Combinatorics.Crossing.CircleOffset
import OAI.Combinatorics.Crossing.CircleDeterminantOrder
import Mathlib.Tactic.FunProp
namespace OAI
noncomputable section
namespace Zarankiewicz
open scoped Topology
def offsetParameter (a δ : ℝ) : ℝ :=
(circleOffset (circlePoint a) δ).2/(1+(circleOffset (circlePoint a) δ).1)
theorem continuous_circleOffset (u : Plane) : Continuous (circleOffset u) := by
change Continuous (fun δ : ℝ => Real.sqrt (1-δ^2) • u+δ • rotateQuarter u)
fun_prop
@[simp] theorem offsetParameter_zero (a : ℝ) : offsetParameter a 0=a := by
simp only [offsetParameter,circleOffset_zero,circlePoint_parameter]
theorem continuousAt_offsetParameter_zero (a : ℝ) : ContinuousAt (offsetParameter a) 0 := by
apply ContinuousAt.div
· exact (continuous_circleOffset (circlePoint a)).continuousAt.snd
· exact continuousAt_const.add (continuous_circleOffset (circlePoint a)).continuousAt.fst
· rw [circleOffset_zero,circlePoint_one_add_x]
exact div_ne_zero (by norm_num) (ne_of_gt (circle_denominator_pos a))
theorem exists_circle_offset_parameter_radius (a ε : ℝ) (hε : 0<ε) :
∃ ρ : ℝ, 0<ρ ∧ ∀ δ : ℝ, |δ|<ρ →
circlePoint (offsetParameter a δ)=circleOffset (circlePoint a) δ ∧
|offsetParameter a δ-a|<ε ∧
(0<δ → a<offsetParameter a δ) ∧ (δ<0 → offsetParameter a δ<a) := by
have hp := continuousAt_offsetParameter_zero a
have hnear : ∀ᶠ δ : ℝ in 𝓝 0, offsetParameter a δ ∈ Set.Ioo (a-ε) (a+ε) :=
hp.eventually (isOpen_Ioo.mem_nhds (by rw [offsetParameter_zero]; constructor <;> linarith))
have hsmall : ∀ᶠ δ : ℝ in 𝓝 0, |δ|<1 := by
apply Metric.eventually_nhds_iff.mpr
exact ⟨1,by norm_num,by intro delta; simp [Real.dist_eq]⟩
have hcut : ∀ᶠ δ : ℝ in 𝓝 0, (circleOffset (circlePoint a) δ).1 ≠ -1 :=
(continuous_circleOffset (circlePoint a)).continuousAt.fst.eventually_ne
(by simpa using circlePoint_avoids_cut a)
have hprodcont : ContinuousAt (fun δ : ℝ => 1+a*offsetParameter a δ) 0 :=
continuousAt_const.add (continuousAt_const.mul hp)
have hprod : ∀ᶠ δ : ℝ in 𝓝 0, 0<1+a*offsetParameter a δ :=
continuousAt_const.eventually_lt hprodcont (by rw [offsetParameter_zero]; nlinarith [sq_nonneg a])
have hevent : ∀ᶠ δ : ℝ in 𝓝 0,
circlePoint (offsetParameter a δ)=circleOffset (circlePoint a) δ ∧
|offsetParameter a δ-a|<ε ∧
(0<δ → a<offsetParameter a δ) ∧ (δ<0 → offsetParameter a δ<a) := by
filter_upwards [hnear,hsmall,hcut,hprod] with δ hnear hsmall hcut hprod
have hunit : squaredLength (circlePoint a)=1 := circlePoint_on_unit_circle a
have he := circlePoint_surjective_away_cut (circleOffset (circlePoint a) δ)
(circleOffset_on_circle _ hunit δ hsmall.le) hcut
change circlePoint (offsetParameter a δ)=circleOffset (circlePoint a) δ at he
have hdet : det2 (circlePoint a) (circlePoint (offsetParameter a δ))=δ := by
rw [he]
exact circleOffset_side _ hunit δ
refine ⟨he,?_,?_,?_⟩
· exact abs_lt.mpr ⟨by linarith [hnear.1],by linarith [hnear.2]⟩
· intro hδ
apply (circle_det_pos_iff a _ hprod).mp
rwa [hdet]
· intro hδ
apply (circle_det_neg_iff a _ hprod).mp
rwa [hdet]
obtain ⟨ρ,hρ,hh⟩ := Metric.eventually_nhds_iff.mp hevent
exact ⟨ρ,hρ,fun δ hδ => hh (by simpa [Real.dist_eq] using hδ)⟩
end Zarankiewicz
end
end OAI
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