openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalFreeFirstLimit.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalColumnLimit | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralFreeSlowIdentification | |
| namespace OAI | |
| /-! The canonical first column converges to the explicit free angular H column. -/ | |
| open Filter Topology Set Polynomial | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem canonical_circular_first_H_limit | |
| (ν m νp νm : ℕ → ℂ) (ν₀ m₀ νm₀ q : ℂ) (ell : ℕ) | |
| (hν : Tendsto ν atTop (𝓝 ν₀)) (hm : Tendsto m atTop (𝓝 m₀)) | |
| (hp : Tendsto νp atTop (𝓝 ((ell : ℂ)-2*q))) | |
| (hn : Tendsto νm atTop (𝓝 νm₀)) (hq : -1 < q.re) | |
| (δ L : ℝ) (hδ : 0 < δ) (hsmall : ‖m₀‖+2*δ < 1) | |
| (hX : ∀ᶠ n in atTop, HasRadialExterior (ν n) n (m n) L) | |
| (W : ℕ → ℝ → E₄) | |
| (hW : ∀ᶠ n in atTop, ∀ t, 0 ≤ t → HasDerivAt (W n) | |
| (circularLeadingField t (W n t)+circularBoundedField (νp n) (νm n) | |
| ((ell*(ell+10) : ℕ) : ℂ) n (radialExteriorCanonical (ν n) n (m n) L t).1 | |
| (W n t)) t) | |
| (heW : ∀ᶠ n in atTop, ∀ J : ℕ, ∃ k : ℕ, J ≤ k ∧ ∃ u : CircularTailSpace, | |
| ∀ t, 0 ≤ t → W n t=circularPolynomialJet | |
| (spectralOutgoingPolynomial (νp n) (νm n) ((ell*(ell+10) : ℕ) : ℂ) n | |
| (radialExteriorExpansion (ν n) n (m n) k) (1,0) k) t+ | |
| circularUnweight (2*(k : ℝ)) u t) : | |
| ∃ T : ℝ, 0 ≤ T ∧ L ≤ T ∧ | |
| TendstoUniformlyOn W (spectralFreeFirstColumn ell q) atTop (Ici T) := by | |
| let η : ℂ := ((ell*(ell+10) : ℕ) : ℂ) | |
| let νp₀ : ℂ := (ell : ℂ)-2*q | |
| obtain ⟨j,hj⟩ := exists_nat_gt (max (radialExteriorMatrixBound ν₀) | |
| (circularFieldBound νp₀ νm₀ η 1 0)) | |
| have hj₁ : radialExteriorMatrixBound ν₀ < 2*((j+1 : ℕ) : ℝ) := by | |
| have hh := (le_max_left (radialExteriorMatrixBound ν₀) | |
| (circularFieldBound νp₀ νm₀ η 1 0)).trans_lt hj | |
| push_cast | |
| linarith [Nat.cast_nonneg (α := ℝ) j] | |
| have hj₂ : circularFieldBound νp₀ νm₀ η 1 0 < 2*((j+1 : ℕ) : ℝ) := by | |
| have hh := (le_max_right (radialExteriorMatrixBound ν₀) | |
| (circularFieldBound νp₀ νm₀ η 1 0)).trans_lt hj | |
| push_cast | |
| linarith [Nat.cast_nonneg (α := ℝ) j] | |
| obtain ⟨T,ρ,hT,hLT,_hρ,_hρ1,v,w,_hvw,hlim,_hlimn,_hlim₀,hactual,hfree⟩ := | |
| exists_canonical_circular_solution_limit ν m νp νm ν₀ m₀ νp₀ νm₀ η | |
| hν hm hp hn δ L hδ hsmall hX (j+1) hj₁ hj₂ (1,0) | |
| let U := fun n => spectralOutgoingPolynomial (νp n) (νm n) η n | |
| (radialExteriorExpansion (ν n) n (m n) (j+1)) (1,0) (j+1) | |
| let Y := fun n t => circularPolynomialJet (U n) t+ | |
| circularUnweight (2*((j+1 : ℕ) : ℝ)) (v n) t | |
| let Y₀ := fun t => circularPolynomialJet | |
| (spectralOutgoingPolynomial νp₀ νm₀ η 1 0 (1,0) (j+1)) t+ | |
| circularUnweight (2*((j+1 : ℕ) : ℝ)) w t | |
| have heq : ∀ᶠ n in atTop, ∀ t, T ≤ t → Y n t=W n t := by | |
| filter_upwards [hactual,hW,heW,eventually_ge_atTop 1] with n han hwn hen hnn | |
| exact circular_expansion_matches_allOrders (ν n) (νp n) (νm n) η (m n) n hnn | |
| (1,0) (j+1) (v n) ρ T hT | |
| (fun t => (radialExteriorCanonical (ν n) n (m n) L t).1) (Y n) (W n) | |
| (fun t ht => (han.2 t ht).1) (fun t ht => (han.2 t ht).2) | |
| (fun t ht => hwn t (hT.trans ht)) (fun _ _ => rfl) hen han.1 | |
| let S := max T (max 0 (Real.log 4/2)) | |
| have hH : ∀ t, S ≤ t → Y₀ t=spectralFreeFirstColumn ell q t := | |
| spectralFreeFirstColumn_eq_correction ell q νm₀ hq j w T hj₂ hfree | |
| refine ⟨S,(le_max_left 0 _).trans (le_max_right T _), | |
| hLT.trans (le_max_left T _),?_⟩ | |
| rw [Metric.tendstoUniformlyOn_iff] | |
| intro ε hε | |
| filter_upwards [(Metric.tendstoUniformlyOn_iff.mp hlim) ε hε,heq] with n hn he t ht | |
| have htT : T ≤ t := (le_max_left T _).trans ht | |
| rw [← he t htT,← hH t ht] | |
| exact hn t (hT.trans htT) | |
| end DefocusingNLS | |
| end OAI | |