| import OAI.Geometry.CylinderCovering.Singular.ParameterRangeDefinitions |
| import OAI.Geometry.RuledCovering.Main |
| import OAI.Geometry.CylinderCovering.Singular.Cutoff |
|
|
| namespace OAI |
|
|
| open Filter Topology |
|
|
| namespace CylinderCovering.SingularParameterRange |
| open RuledApproximation |
|
|
| lemma critical_exponent_bounds : |
| 1 < (Real.sqrt 51 - 5) / 2 ∧ (Real.sqrt 51 - 5) / 2 < 2 := by |
| have hs := Real.sq_sqrt (show (0 : ℝ) ≤ 51 by norm_num) |
| have hn := Real.sqrt_nonneg (51 : ℝ) |
| constructor <;> nlinarith |
|
|
| lemma numerator_negative_iff {p : ℝ} (hp : 0 < p) : |
| 2 * p ^ 2 + 10 * p - 13 < 0 ↔ p < (Real.sqrt 51 - 5) / 2 := by |
| have hs := Real.sq_sqrt (show (0 : ℝ) ≤ 51 by norm_num) |
| have hn := Real.sqrt_nonneg (51 : ℝ) |
| constructor |
| · intro h |
| by_contra hnot |
| have hge : Real.sqrt 51 ≤ 2 * p + 5 := by linarith |
| nlinarith |
| · intro h |
| have hlt : 2 * p + 5 < Real.sqrt 51 := by linarith |
| nlinarith |
|
|
| theorem coefficient_negative_iff {p : ℝ} (hp : 1 < p) : |
| singularCoefficient p < 0 ↔ p < (Real.sqrt 51 - 5) / 2 := by |
| have hp0 : 0 < p := by linarith |
| have he : singularCoefficient p * (480 * p) = 2 * p ^ 2 + 10 * p - 13 := by |
| unfold singularCoefficient |
| field_simp |
| ring |
| constructor |
| · intro h |
| apply (numerator_negative_iff hp0).mp |
| rw [← he] |
| exact mul_neg_of_neg_of_pos h (by positivity) |
| · intro h |
| have hn := (numerator_negative_iff hp0).mpr h |
| have hd : 0 < 30 * p := by positivity |
| exact div_neg_of_neg_of_pos (mul_neg_of_pos_of_neg (by norm_num) hn) hd |
|
|
| lemma coefficient_at_critical : |
| singularCoefficient ((Real.sqrt 51 - 5) / 2) = 0 := by |
| have hs := Real.sq_sqrt (show (0 : ℝ) ≤ 51 by norm_num) |
| have hn : 2 * ((Real.sqrt 51 - 5) / 2) ^ 2 + |
| 10 * ((Real.sqrt 51 - 5) / 2) - 13 = 0 := by nlinarith |
| simp [singularCoefficient, hn] |
|
|
| lemma coefficient_at_one : singularCoefficient 1 = -(1 / 480 : ℝ) := by |
| norm_num [singularCoefficient] |
|
|
| theorem coefficient_limit_at_one : |
| Tendsto singularCoefficient (𝓝 1) (𝓝 (-(1 / 480 : ℝ))) := by |
| have hc : ContinuousAt singularCoefficient 1 := by |
| unfold singularCoefficient |
| fun_prop (disch := norm_num) |
| simpa only [coefficient_at_one] using hc.tendsto |
|
|
| theorem cutoff_window_nonempty_iff (p : ℝ) : |
| (∃ γ : ℝ, 1 < γ ∧ γ < 1 / (2 - p)) ↔ 1 < p ∧ p < 2 := by |
| constructor |
| · rintro ⟨γ, hγ, hγp⟩ |
| have hpos : 0 < 1 / (2 - p) := by linarith |
| have hd : 0 < 2 - p := (one_div_pos.mp hpos) |
| have hmul := (lt_div_iff₀ hd).mp hγp |
| constructor |
| · nlinarith |
| · linarith |
| · rintro ⟨hp, hp2⟩ |
| have hd : 0 < 2 - p := by linarith |
| have hlt : 1 < 1 / (2 - p) := (lt_div_iff₀ hd).mpr (by linarith) |
| exact exists_between hlt |
|
|
| theorem endpoint_derivative_determinant {z : Plane} (hz : 0 < z 1) : |
| LinearMap.det (fderiv ℝ (singularField 1) z).toLinearMap = 0 := by |
| simpa using (singularField_trace_det 1 hz).2 |
|
|
| theorem endpoint_derivative_squareZero {z : Plane} (hz : 0 < z 1) : |
| (fderiv ℝ (singularField 1) z).comp (fderiv ℝ (singularField 1) z) = 0 := by |
| rw [singularField_fderiv 1 hz] |
| ext w i |
| fin_cases i <;> |
| simp [ContinuousLinearMap.comp_apply, matrixCLM_apply, vec2] <;> ring |
|
|
| theorem endpoint_eigenvalue_eq_zero {z : Plane} (hz : 0 < z 1) {μ : ℝ} |
| (hμ : Module.End.HasEigenvalue (fderiv ℝ (singularField 1) z).toLinearMap μ) : |
| μ = 0 := by |
| obtain ⟨u, hu⟩ := hμ.exists_hasEigenvector |
| have he : (fderiv ℝ (singularField 1) z) |
| ((fderiv ℝ (singularField 1) z) u) = 0 := |
| congrArg (fun A : Plane →L[ℝ] Plane => A u) (endpoint_derivative_squareZero hz) |
| change (fderiv ℝ (singularField 1) z).toLinearMap |
| ((fderiv ℝ (singularField 1) z).toLinearMap u) = 0 at he |
| rw [hu.apply_eq_smul, map_smul, hu.apply_eq_smul, smul_smul] at he |
| have hm : μ * μ = 0 := (smul_eq_zero.mp he).resolve_right hu.2 |
| exact mul_self_eq_zero.mp hm |
|
|
| theorem statement : Statement where |
| negative_range _ hp := coefficient_negative_iff hp |
| critical_zero := coefficient_at_critical |
| endpoint_value := coefficient_at_one |
| endpoint_limit := coefficient_limit_at_one |
| cutoff_window := cutoff_window_nonempty_iff |
| endpoint_degenerate _ hz := |
| ⟨endpoint_derivative_determinant hz, fun _ hμ => endpoint_eigenvalue_eq_zero hz hμ⟩ |
|
|
| end CylinderCovering.SingularParameterRange |
|
|
| end OAI |
|
|
