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785c727 | 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | 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
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