| 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 | |