File size: 3,219 Bytes
fac6777 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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
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