openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalParameterLogJets.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralParameterLogJets | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalParameterEquation | |
| namespace OAI | |
| /-! The actual canonical parameter derivative has bounded normalized jets. -/ | |
| open Set Filter Topology | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem canonical_circular_parameter_logJets (ν η b : ℂ) (n : ℕ) (hn : 1≤n) | |
| (L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0) | |
| (c : ℂ × ℂ) (Y : ℂ → ℝ → E₄) | |
| (hY : IsCanonicalHolomorphicColumn ν η b n L c Y) (z : ℂ) : | |
| HasLogJetBound 0 (fun t => (deriv (fun lam => Y lam t) z).1.1) ∧ | |
| HasLogJetBound 0 (fun t => (deriv (fun lam => Y lam t) z).2.1) := by | |
| let q := radialExteriorCanonical ν n b L | |
| let D := fun t => deriv (fun lam => Y lam t) z | |
| let νp := ν-2*z | |
| let νm := star ν-2*z | |
| let L₀ := max L 0 | |
| have hq (t : ℝ) (ht : L₀<t) : HasDerivAt q (radialExteriorODEField ν n t (q t)) t := | |
| ((radialExteriorCanonical_spec hX).2.2 t ((le_max_left L 0).trans ht.le)).2 | |
| have hYe (t : ℝ) (ht : L₀<t) : HasDerivAt (Y z) | |
| (circularLeadingField t (Y z t)+circularBoundedField νp νm η n (q t).1 (Y z t)) t := | |
| hY.1 z t ((le_max_right L 0).trans ht.le) | |
| have hDe (t : ℝ) (ht : L₀<t) : HasDerivAt D | |
| (circularLeadingField t (D t)+circularBoundedField νp νm η n (q t).1 (D t)+ | |
| circularPointSlopeCLM νp νm 0 (Y z t)) t := by | |
| have h := canonical_circular_parameter_equation ν η b n hn L hX hb c Y hY z t | |
| ((le_max_right L 0).trans_lt ht) | |
| rw [circularPointSlope_shift] at h | |
| exact h | |
| let Mq := ‖(b,(0 : ℂ))‖+1 | |
| let MY := ‖((c.1,(0 : ℂ)),(c.2,(0 : ℂ)))‖+1 | |
| obtain ⟨Uq,hUq⟩ := eventually_atTop.mp | |
| ((radialExteriorCanonical_spec hX).2.1.tendsto.norm.eventually | |
| (gt_mem_nhds (show ‖(b,(0 : ℂ))‖<Mq by dsimp only [Mq]; linarith))) | |
| obtain ⟨UY,hUY⟩ := eventually_atTop.mp | |
| ((hY.2.1 z).norm.eventually | |
| (gt_mem_nhds (show ‖((c.1,(0 : ℂ)),(c.2,(0 : ℂ)))‖<MY by dsimp only [MY]; linarith))) | |
| obtain ⟨C,hC⟩ := canonical_circular_parameter_bounded ν η b n hn L hX hb c Y hY z | |
| let M := max Mq (max MY C) | |
| let U := max 0 (max Uq UY) | |
| have hM : 0≤M := (by dsimp only [Mq]; positivity : 0≤Mq).trans (le_max_left _ _) | |
| have hbq (t : ℝ) (ht : U≤t) : ‖q t‖≤M := | |
| (hUq t (((le_max_left Uq UY).trans (le_max_right 0 _)).trans ht)).le.trans (le_max_left _ _) | |
| have hbY (t : ℝ) (ht : U≤t) : ‖Y z t‖≤M := | |
| (hUY t (((le_max_right Uq UY).trans (le_max_right 0 _)).trans ht)).le.trans | |
| ((le_max_left MY C).trans (le_max_right _ _)) | |
| have hbD (t : ℝ) (ht : U≤t) : ‖D t‖≤M := | |
| (hC t ((le_max_left 0 _).trans ht)).trans | |
| ((le_max_right MY C).trans (le_max_right _ _)) | |
| obtain ⟨P,hP⟩ := canonical_parameter_polynomial_approximation ν η b n hn L hX hb c Y hY z | |
| have hpos := spectralParameter_positive_derivatives ν νp νm η n q (Y z) D L₀ U M hM | |
| hq hYe hDe hbq hbY hbD P (by | |
| intro J | |
| obtain ⟨A,hA,hAT⟩ := hP J | |
| exact ⟨A,0,hA,hAT⟩) | |
| have hs := spectralParameter_solution_contDiffOn ν νp νm η n q (Y z) D L₀ hq hYe hDe | |
| exact ⟨HasLogJetBound.of_bounded_positive_decay _ L₀ U M hM hs.fst.fst | |
| (fun t ht => (norm_fst_le _).trans ((norm_fst_le _).trans (hbD t ht))) hpos.1, | |
| HasLogJetBound.of_bounded_positive_decay _ L₀ U M hM hs.snd.fst | |
| (fun t ht => (norm_fst_le _).trans ((norm_snd_le _).trans (hbD t ht))) hpos.2⟩ | |
| end DefocusingNLS | |
| end OAI | |