openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Certificates /AnalyticMatchingColumn.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Certificates.NormalizedMatchingColumns | |
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.SlowParameterHolomorphy | |
| namespace OAI | |
| /-! # The analytic boundary column and its normalized tail representation -/ | |
| open Matrix | |
| namespace DefocusingNLS | |
| /-- The derivative-shift formula gives an explicitly analytic boundary column. -/ | |
| noncomputable def slowBoundaryColumn (q : ℂ) (m : ℕ) (s : ℂ) : Fin 2 → ℂ := | |
| ![regularizedSlowSolution q m (-s), q * regularizedSlowSolution (q + 1) (m + 1) (-s)] | |
| theorem slowBoundaryColumn_eq_jet (q : ℂ) (m : ℕ) (s : ℂ) | |
| (hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) : | |
| slowBoundaryColumn q m s = | |
| ![regularizedSlowSolution q m (-s), -deriv (regularizedSlowSolution q m) (-s)] := by | |
| have hs0 : -s ≠ 0 := neg_ne_zero.mpr (fun h => hsim (by simp [h])) | |
| have hd := (hasDerivAt_regularizedSlowSolution_shift_closed q m (-s) hq | |
| (by simp [hsre]) hs0).deriv | |
| simp [slowBoundaryColumn, hd] | |
| theorem differentiableAt_slowBoundaryColumn (q : ℂ) (m : ℕ) (s : ℂ) | |
| (hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) : | |
| DifferentiableAt ℂ (fun z => slowBoundaryColumn z m s) q := by | |
| have hs0 : -s ≠ 0 := neg_ne_zero.mpr (fun h => hsim (by simp [h])) | |
| have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith | |
| have h₀ := differentiableAt_regularizedSlowSolution_parameter_closed q m (-s) hq | |
| (by simp [hsre]) hs0 | |
| have h₁ := differentiableAt_regularizedSlowSolution_parameter_closed (q + 1) (m + 1) | |
| (-s) hq' (by simp [hsre]) hs0 | |
| apply differentiableAt_pi.mpr | |
| intro i | |
| fin_cases i | |
| · change DifferentiableAt ℂ (fun z => regularizedSlowSolution z m (-s)) q | |
| exact h₀ | |
| · change DifferentiableAt ℂ (fun z => z * regularizedSlowSolution (z + 1) (m + 1) (-s)) q | |
| convert! differentiableAt_id.mul (h₁.comp q (differentiableAt_id.add_const 1)) using 1 | |
| theorem analyticOnNhd_slowBoundaryColumn (m : ℕ) (s : ℂ) | |
| (hsre : s.re = 0) (hsim : s.im ≠ 0) : | |
| AnalyticOnNhd ℂ (fun q => slowBoundaryColumn q m s) {q : ℂ | -1 < q.re} := by | |
| apply DifferentiableOn.analyticOnNhd _ (isOpen_lt continuous_const Complex.continuous_re) | |
| intro q hq | |
| exact (differentiableAt_slowBoundaryColumn q m s hq hsre hsim).differentiableWithinAt | |
| theorem slowBoundaryColumn_ne_zero (q : ℂ) (m : ℕ) (s : ℂ) | |
| (hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) : | |
| slowBoundaryColumn q m s ≠ 0 := by | |
| rw [slowBoundaryColumn_eq_jet q m s hq hsre hsim] | |
| intro hz | |
| have h₀ := congrFun hz 0 | |
| have h₁ := congrFun hz 1 | |
| simp only [Matrix.cons_val_zero, Matrix.cons_val_one, Pi.zero_apply] at h₀ h₁ | |
| have hs0 : -s ≠ 0 := neg_ne_zero.mpr (fun h => hsim (by simp [h])) | |
| exact (regularizedSlowSolution_jet_nonzero q m (-s) hq (by simp [hsre]) hs0).elim | |
| (fun h => h h₀) (fun h => h (neg_eq_zero.mp h₁)) | |
| theorem normalizedMatching_tail_disk (σ : ℝ) (ℓ K : ℕ) (q s : ℂ) | |
| (hσ : -(1 / 32 : ℝ) ≤ σ) (hK : 3 ≤ K) | |
| (hq : q.re = σ + (ℓ : ℝ) / 2) (hsre : s.re = 0) (hsim : s.im ≠ 0) : | |
| normalizedMatchingC q (ℓ + 6) s K ≠ 0 ∧ | |
| Complex.normSq (normalizedMatchingB q (ℓ + 6) s K / | |
| normalizedMatchingC q (ℓ + 6) s K + s / ((ℓ : ℝ) + 5 : ℂ)) < | |
| Complex.normSq (s / ((ℓ : ℝ) + 5 : ℂ)) := by | |
| have hq' : -1 < q.re := by rw [hq]; linarith [Nat.cast_nonneg (α := ℝ) ℓ] | |
| have hp := shiftedPochhammer_ne_zero q (K - 1) hq' | |
| obtain ⟨hC, hd⟩ := normalizedSlow_tail_disk σ ℓ K q s hσ hK hq hsre hsim | |
| refine ⟨div_ne_zero hC hp, ?_⟩ | |
| have he : normalizedMatchingB q (ℓ + 6) s K / normalizedMatchingC q (ℓ + 6) s K = | |
| normalizedSlowB q (ℓ + 6) s K / normalizedSlowC q (ℓ + 6) s K := by | |
| unfold normalizedMatchingB normalizedMatchingC | |
| field_simp | |
| rwa [he] | |
| /-- The backward image of the ratio column differs from the true analytic | |
| boundary column by the explicit nonzero tail scalar. -/ | |
| theorem normalizedMatching_ratio_column (q : ℂ) (M K : ℕ) (s : ℂ) | |
| (hK : 0 < K) (hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) | |
| (hC : normalizedMatchingC q (M + 1) s K ≠ 0) : | |
| backwardProduct M s q K *ᵥ | |
| ![normalizedMatchingB q (M + 1) s K / normalizedMatchingC q (M + 1) s K, 1] = | |
| (normalizedMatchingC q (M + 1) s K)⁻¹ • slowBoundaryColumn q (M + 1) s := by | |
| have hv : ![normalizedMatchingB q (M + 1) s K / normalizedMatchingC q (M + 1) s K, 1] = | |
| (normalizedMatchingC q (M + 1) s K)⁻¹ • | |
| ![normalizedMatchingB q (M + 1) s K, normalizedMatchingC q (M + 1) s K] := by | |
| ext i | |
| fin_cases i <;> simp [div_eq_mul_inv, mul_comm, hC] | |
| rw [hv, Matrix.mulVec_smul, normalizedMatching_backwardProduct q M K s hK hq hsre hsim, | |
| slowBoundaryColumn_eq_jet q (M + 1) s hq hsre hsim] | |
| end DefocusingNLS | |
| end OAI | |