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ddc84c5 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | import OAI.Analysis.DirectCrouzeix.Convexity
namespace OAI
universe u_143
noncomputable section
open scoped Matrix Matrix.Norms.L2Operator Kronecker
noncomputable section
open MeasureTheory Set Filter Metric
open scoped Topology Interval ENNReal NNReal ComplexConjugate
noncomputable section
open Filter Metric Set
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set Filter Metric
open scoped Topology ComplexConjugate
noncomputable section
open Set
open scoped ComplexConjugate Matrix
namespace DirectCrouzeix.Geometry
theorem positive_parallel_of_complex_inner {G q : ℂ} (hG : G ≠ 0) (hq : q ≠ 0)
(horth : inner ℝ G (Complex.I*q) = 0) (hpos : 0 ≤ inner ℝ G q) :
∃ t : ℝ, 0 < t ∧ q = t • G := by
have him : (conj G*q).im = 0 := by
rw [real_inner_eq_re_inner ℂ,RCLike.inner_apply] at horth
change ((Complex.I*q)*conj G).re = 0 at horth
simp only [Complex.mul_re,Complex.mul_im,Complex.I_re,Complex.I_im,Complex.conj_re,Complex.conj_im] at horth ⊢
nlinarith
have hre : (conj G*q).re = inner ℝ G q := by
simp [real_inner_eq_re_inner ℂ,RCLike.inner_apply,mul_comm]
have hnon : inner ℝ G q ≠ 0 := by
intro hh
have he : conj G*q = 0 := Complex.ext (by simpa [hre] using hh) (by simpa using him)
exact mul_ne_zero ((map_ne_zero (starRingEnd ℂ)).mpr hG) hq he
have hp : 0 < inner ℝ G q := lt_of_le_of_ne hpos (Ne.symm hnon)
let t := inner ℝ G q/‖G‖^2
refine ⟨t,div_pos hp (sq_pos_of_pos (norm_pos_iff.mpr hG)),?_⟩
have hprod : conj G*q = (inner ℝ G q:ℂ) := by
apply Complex.ext <;> simp [hre,him]
have hGG : conj G*G = ((‖G‖^2:ℝ):ℂ) := by
rw [← Complex.normSq_eq_conj_mul_self,Complex.normSq_eq_norm_sq]
apply mul_left_cancel₀ ((map_ne_zero (starRingEnd ℂ)).mpr hG : conj G ≠ 0)
rw [hprod,Complex.real_smul,mul_left_comm,hGG]
dsimp only [t]
rw [Complex.ofReal_div,Complex.ofReal_pow,div_mul_cancel₀ _ (pow_ne_zero _ (Complex.ofReal_ne_zero.mpr (norm_ne_zero_iff.mpr hG)))]
theorem expLevel_holomorphic_support {ι : Type u_143} [Fintype ι]
(v : ι → ℂ) (b : ι → ℝ) {a : ℂ} (ha : expLevel v b a < 1)
{ψ : ℂ → ℂ} {ζ : ℂ} (hζ : ‖ζ‖ = 1)
(hd : HasDerivAt ψ (deriv ψ ζ) ζ) (hdn : deriv ψ ζ ≠ 0)
(hcircle : ∀ η : ℂ, ‖η‖ = 1 → expLevel v b (ψ η) = 1)
(hout : ∀ η : ℂ, 1 ≤ ‖η‖ → 1 ≤ expLevel v b (ψ η))
{w : ℂ} (hw : expLevel v b w ≤ 1) :
0 ≤ inner ℝ (ζ*deriv ψ ζ) (ψ ζ-w) := by
let G := expGradient v b (ψ ζ)
let q := ζ*deriv ψ ζ
have hζ0 : ζ ≠ 0 := by intro hh; simp [hh] at hζ
have hq : q ≠ 0 := mul_ne_zero hζ0 hdn
have hlevel : expLevel v b (ψ ζ) = 1 := hcircle ζ hζ
have hG : G ≠ 0 := expGradient_ne_zero v b hlevel ha
have hφd : HasFDerivAt (expLevel v b) (innerSL ℝ G) (ψ ζ) := expLevel_hasFDerivAt v b (ψ ζ)
have horth : inner ℝ G (Complex.I*q) = 0 := by
let e := fun t : ℝ => ζ*Complex.exp ((t:ℂ)*Complex.I)
have he0 : e 0 = ζ := by simp [e]
have hed : HasDerivAt e (ζ*Complex.I) 0 := by
have hi : HasDerivAt (fun t : ℝ => (t:ℂ)) 1 0 := by
simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one] using! (Complex.ofRealCLM.hasDerivAt (x := (0:ℝ)))
convert! ((hi.mul_const Complex.I).cexp).const_mul ζ using 1 ; simp
have hψd := (show HasDerivAt ψ (deriv ψ ζ) (e 0) from he0 ▸ hd).scomp 0 hed
have hcomp := (show HasFDerivAt (expLevel v b) (innerSL ℝ G) (ψ (e 0)) from he0 ▸ hφd).comp_hasDerivAt 0 hψd
have hc : (fun t : ℝ => expLevel v b (ψ (e t))) = fun _ => 1 := by
funext t
apply hcircle
simp [e,hζ,Complex.norm_exp]
have hv : deriv (fun t : ℝ => expLevel v b (ψ (e t))) 0 = 0 := by rw [hc]; simp
have heval : inner ℝ G (Complex.I*q) = (innerSL ℝ G) ((ζ*Complex.I) • deriv ψ ζ) := by
simp only [innerSL_apply_apply,smul_eq_mul,q]; congr 1; ring
rw [heval,← hcomp.deriv]
exact hv
have hpos : 0 ≤ inner ℝ G q := by
let r := fun t : ℝ => (t:ℂ)*ζ
have hr1 : r 1 = ζ := by simp [r]
have hrd : HasDerivAt r ζ 1 := by
have hi : HasDerivAt (fun t : ℝ => (t:ℂ)) 1 1 := by
simpa only [Complex.ofRealCLM_apply,Complex.ofReal_one] using! (Complex.ofRealCLM.hasDerivAt (x := (1:ℝ)))
convert! hi.mul_const ζ using 1 ; simp
have hψd := (show HasDerivAt ψ (deriv ψ ζ) (r 1) from hr1 ▸ hd).scomp 1 hrd
have hcomp := (show HasFDerivAt (expLevel v b) (innerSL ℝ G) (ψ (r 1)) from hr1 ▸ hφd).comp_hasDerivAt 1 hψd
have hmin : IsMinOn (fun t : ℝ => expLevel v b (ψ (r t))) (Ici 1) 1 := by
intro t ht
change expLevel v b (ψ (r 1)) ≤ expLevel v b (ψ (r t))
rw [hr1,hlevel]
apply hout
have ht0 : 0 ≤ t := le_trans (by norm_num) ht
simpa [r,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht0,hζ] using ht
have htan : (1:ℝ) ∈ posTangentConeAt (Ici 1) 1 := by
apply mem_posTangentConeAt_of_segment_subset
exact (convex_Ici (1:ℝ)).segment_subset (by norm_num) (by norm_num)
have hh := hmin.isLocalMinOn.hasFDerivWithinAt_nonneg hcomp.hasFDerivAt.hasFDerivWithinAt htan
simpa [q] using hh
obtain ⟨t,ht,hqeq⟩ := positive_parallel_of_complex_inner hG hq horth hpos
have hs := expLevel_first_order v b (ψ ζ) w
rw [hlevel,show expGradient v b (ψ ζ) = G from rfl] at hs
have hsupport : 0 ≤ inner ℝ G (ψ ζ-w) := by
rw [inner_sub_right] at hs ⊢
linarith
change 0 ≤ inner ℝ q (ψ ζ-w)
rw [hqeq,real_inner_smul_left]
exact mul_nonneg (le_of_lt ht) hsupport
end DirectCrouzeix.Geometry
namespace DirectCrouzeix.Geometry
end DirectCrouzeix.Geometry
end
end
end
end
end
end
end
end
end
end OAI
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