| 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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|
