openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileMatchingQuotient.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileNonvanishing | |
| namespace OAI | |
| /-! The actual logarithmic derivative equals the quotient of the matched column. -/ | |
| open Matrix | |
| namespace DefocusingNLS.ProfileCertificate | |
| attribute [local irreducible] backwardProduct profileProduct | |
| normalizedMatchingB normalizedMatchingC | |
| noncomputable def freeProfileJ (b Z : ℝ) : ℂ := | |
| let q : ℂ := -Complex.I*(b : ℂ) | |
| q*regularizedSlowSolution (q+1) 7 (-Complex.I*(Z : ℂ))/ | |
| regularizedSlowSolution q 6 (-Complex.I*(Z : ℂ)) | |
| /-- The normalization of the finite matrix cancels in the component ratio. -/ | |
| theorem freeProfileJ_eq_matched (b z : ℝ) | |
| (hz : |z| ≤ (radius : ℝ)) : | |
| let q : ℂ := -Complex.I*(((centerB : ℝ)+b : ℝ) : ℂ) | |
| let s : ℂ := Complex.I*(((centerZ : ℝ)+z : ℝ) : ℂ) | |
| let r := normalizedMatchingB q 6 s 34/normalizedMatchingC q 6 s 34 | |
| freeProfileJ ((centerB : ℝ)+b) ((centerZ : ℝ)+z) = | |
| (normalizedProfile b z 1 0*r+normalizedProfile b z 1 1)/ | |
| (normalizedProfile b z 0 0*r+normalizedProfile b z 0 1) := by | |
| dsimp only | |
| let q : ℂ := -Complex.I*(((centerB : ℝ)+b : ℝ) : ℂ) | |
| let s : ℂ := Complex.I*(((centerZ : ℝ)+z : ℝ) : ℂ) | |
| let r := normalizedMatchingB q 6 s 34/normalizedMatchingC q 6 s 34 | |
| have hq : -1 < q.re := by simp [q] | |
| have hsre : s.re = 0 := by simp [s] | |
| have hsim : s.im ≠ 0 := by | |
| have he := (abs_le.mp hz).1 | |
| have hZ : 0 < (centerZ : ℝ)+z := by norm_num [centerZ, radius] at he ⊢; linarith | |
| simpa [s] using hZ.ne' | |
| have hC := (profile_tail_disk b z hz).1 | |
| have hv := normalizedMatching_ratio_column q 5 34 s (by decide) hq hsre hsim hC | |
| have h0 := congrFun hv 0 | |
| have h1 := congrFun hv 1 | |
| simp only [Matrix.mulVec, dotProduct, Fin.sum_univ_two, Matrix.cons_val_zero, | |
| Matrix.cons_val_one, mul_one, Pi.smul_apply, smul_eq_mul, slowBoundaryColumn] at h0 h1 | |
| change backwardProduct 5 s q 34 0 0*r+backwardProduct 5 s q 34 0 1 = | |
| (normalizedMatchingC q 6 s 34)⁻¹*regularizedSlowSolution q 6 (-s) at h0 | |
| change backwardProduct 5 s q 34 1 0*r+backwardProduct 5 s q 34 1 1 = | |
| (normalizedMatchingC q 6 s 34)⁻¹*(q*regularizedSlowSolution (q+1) 7 (-s)) at h1 | |
| have hfac : (Nat.factorial 34 : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero 34 | |
| have hn : ∀ i j, normalizedProfile b z i j = | |
| ((Nat.factorial 34 : ℂ)⁻¹/centralM)*backwardProduct 5 s q 34 i j := by | |
| intro i j | |
| unfold normalizedProfile profileProduct | |
| simp only [Matrix.smul_apply, smul_eq_mul] | |
| change ((Nat.factorial 34 : ℂ)⁻¹*backwardProduct 5 s q 34 i j)/centralM = _ | |
| ring | |
| rw [hn 1 0, hn 1 1, hn 0 0, hn 0 1] | |
| simp only [mul_assoc] | |
| rw [← mul_add, ← mul_add] | |
| change freeProfileJ ((centerB : ℝ)+b) ((centerZ : ℝ)+z) = | |
| _*(backwardProduct 5 s q 34 1 0*r+backwardProduct 5 s q 34 1 1)/ | |
| (_*(backwardProduct 5 s q 34 0 0*r+backwardProduct 5 s q 34 0 1)) | |
| rw [h0, h1] | |
| have hscale : ((Nat.factorial 34 : ℂ)⁻¹/centralM) ≠ 0 := | |
| div_ne_zero (inv_ne_zero hfac) centralM_ne_zero | |
| rw [mul_div_mul_left _ _ hscale, mul_div_mul_left _ _ (inv_ne_zero hC)] | |
| simp only [freeProfileJ, q, s, neg_mul] | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |