openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalParameterSmooth.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalParameterPower | |
| namespace OAI | |
| /-! Smoothness of the physical parameter values on a sufficiently large exterior interval. -/ | |
| open Set Filter Topology | |
| open scoped ContDiff | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem canonicalPhysicalParameterValues_smooth_tail (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) : | |
| ∃ R : ℝ, 1<R ∧ ContDiffOn ℝ ∞ (canonicalPhysicalParameterValues ν Y z) (Ioi R) := by | |
| let f := fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).1.1 | |
| let g := fun t => (circularParameterShear t (Y z t) (deriv (fun lam => Y lam t) z)).2.1 | |
| have hj := canonical_parameterShear_value_logJets ν η b n hn L hX hb c Y hY z 1 zero_lt_one | |
| obtain ⟨Lf,hfs⟩ := hj.1.smooth | |
| obtain ⟨Lg,hgs⟩ := hj.2.smooth | |
| let T := max Lf Lg | |
| let R := max (Real.exp T) 1+1 | |
| have hR : 1<R := by dsimp only [R]; linarith [le_max_right (Real.exp T) 1] | |
| have hER : Real.exp T≤R := by dsimp only [R]; linarith [le_max_left (Real.exp T) 1] | |
| have hp := spectralPowerValues_smooth (ν-2*z) (star ν-2*z) f g T | |
| (hfs.mono (Ioi_subset_Ioi (le_max_left _ _))) | |
| (hgs.mono (Ioi_subset_Ioi (le_max_right _ _))) | |
| refine ⟨R,hR,(hp.mono (Ioi_subset_Ioi hER)).congr ?_⟩ | |
| intro r hr | |
| exact canonicalPhysicalParameterValues_eq ν η b n L c Y hY z r (hR.trans hr) | |
| end DefocusingNLS | |
| end OAI | |