openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalParameterEquation.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralDifferentiatedOutgoing | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralHolomorphicSingularLimit | |
| namespace OAI | |
| /-! The canonical all-orders column inherits the differentiated equation from | |
| the local weighted-resolvent construction, by outgoing uniqueness. -/ | |
| open Filter Topology | |
| open scoped BoundedContinuousFunction | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem canonical_circular_parameter_equation (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) | |
| (t : ℝ) (ht : 0 < t) : | |
| HasDerivAt (fun s => deriv (fun lam => Y lam s) z) | |
| (circularLeadingField t (deriv (fun lam => Y lam t) z) + | |
| circularBoundedField (ν - 2 * z) (star ν - 2 * z) η n | |
| (radialExteriorCanonical ν n b L t).1 (deriv (fun lam => Y lam t) z) + | |
| circularPointSlopeCLM ν (star ν) z (Y z t)) t := by | |
| obtain ⟨q, hq, hP⟩ := canonical_circular_expansion_data ν b n L hX hb | |
| let C := circularFieldBound (ν - 2 * z) (star ν - 2 * z) η n ‖q‖ | |
| have hC : 0 ≤ C := circularFieldBound_nonneg _ _ _ _ _ | |
| obtain ⟨N, hN⟩ := exists_nat_gt (C + 1) | |
| obtain ⟨j, hNj, e, he⟩ := hP N | |
| have hNj' : (N : ℝ) ≤ j := by exact_mod_cast hNj | |
| have hj : 0 < j := by | |
| have hj' : (0 : ℝ) < j := by linarith | |
| exact_mod_cast hj' | |
| have hgap : C < 2 * (j : ℝ) := by linarith | |
| obtain ⟨W, _hWa, hWd, _hWlim, hWe, hWp⟩ := exists_differentiated_circular_outgoing | |
| ν (star ν) η n hn (radialExteriorExpansion ν n b j) c j hj q e he | |
| have hcont : Continuous (fun lam : ℂ => | |
| circularFieldBound (ν - 2 * lam) (star ν - 2 * lam) η n ‖q‖) := by | |
| unfold circularFieldBound | |
| fun_prop | |
| have hnear : ∀ᶠ lam in 𝓝 z, | |
| circularFieldBound (ν - 2 * lam) (star ν - 2 * lam) η n ‖q‖ < 2 * (j : ℝ) := | |
| hcont.continuousAt.eventually (gt_mem_nhds hgap) | |
| have hsame (s : ℝ) (hs : 0 ≤ s) : | |
| (fun lam => Y lam s) =ᶠ[𝓝 z] (fun lam => W lam s) := by | |
| filter_upwards [hnear] with lam hlam | |
| obtain ⟨k, hjk, w, hw⟩ := hY.2.2.2 lam j | |
| obtain ⟨v, hv⟩ := hWe lam | |
| have heq := circular_outgoing_orders_unique ν (ν - 2 * lam) (star ν - 2 * lam) | |
| η b n hn c j k hjk v w ‖q‖ 0 q (W lam) (Y lam) | |
| (fun r _ => q.norm_coe_le_norm r) (hWd lam hlam) | |
| (fun r hr => by simpa only [hq r hr] using hY.1 lam r hr) | |
| (fun r _ => hv r) hw hlam | |
| exact (heq s (by simpa only [max_self] using hs)).symm | |
| have hderiv (s : ℝ) (hs : 0 ≤ s) : | |
| deriv (fun lam => Y lam s) z = deriv (fun lam => W lam s) z := | |
| (hsame s hs).deriv_eq | |
| have htime : (fun s => deriv (fun lam => Y lam s) z) =ᶠ[𝓝 t] | |
| (fun s => deriv (fun lam => W lam s) z) := by | |
| filter_upwards [eventually_gt_nhds ht] with s hs | |
| exact hderiv s hs.le | |
| have hd := (hWp z hgap t ht.le).congr_of_eventuallyEq htime | |
| apply hd.congr_deriv | |
| rw [hderiv t ht.le, (hsame t ht.le).eq_of_nhds, ← hq t ht.le] | |
| end DefocusingNLS | |
| end OAI | |