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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 71 72 73 74 75 76 77 78 79 80 81 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileCertificateRecipe
import OAI.MathematicalPhysics.DefocusingNLS.Certificates.ForwardMatching
namespace OAI
/-! The rational profile product is the actual normalized backward matrix. -/
open Matrix
namespace DefocusingNLS.ProfileCertificate
noncomputable def complexMatrix (A : QMatrix) : Matrix (Fin 2) (Fin 2) ℂ :=
(Matrix2.toMatrix A).map RationalComplex.toComplex
@[simp] theorem complexMatrix_one : complexMatrix Matrix2.one = 1 := by
ext i j; fin_cases i <;> fin_cases j <;> simp [complexMatrix, Matrix2.one, Matrix2.toMatrix]
@[simp] theorem complexMatrix_mul (A B : QMatrix) :
complexMatrix (Matrix2.mul A B) = complexMatrix A * complexMatrix B := by
simp only [complexMatrix, Matrix2.toMatrix_mul, Matrix.map_mul]
@[simp] theorem complexMatrix_scale (r : RationalComplex) (A : QMatrix) :
complexMatrix (Matrix2.scale r A) = RationalComplex.toComplex r • complexMatrix A := by
simp only [complexMatrix, Matrix2.toMatrix_scale]
exact Matrix.map_smul' _ _ _ RationalComplex.toComplex.map_mul
/-- Each factor in the finite product, including its manuscript normalization. -/
noncomputable def profileFactor (b Z : ℝ) (n : ℕ) : Matrix (Fin 2) (Fin 2) ℂ :=
((n : ℂ)+1)⁻¹ • backwardMatrix 5 (Complex.I * Z) ((n : ℂ)-Complex.I*b)
noncomputable def profileProduct (b Z : ℝ) (K : ℕ) : Matrix (Fin 2) (Fin 2) ℂ :=
((K.factorial : ℂ)⁻¹) • backwardProduct 5 (Complex.I * Z) (-Complex.I*b) K
private theorem factor_complex (n : ℕ) :
complexMatrix (factor n) =
backwardMatrix 5 (Complex.I * (centerZ : ℝ)) ((n : ℂ)-Complex.I*(centerB : ℝ)) := by
ext i j; fin_cases i <;> fin_cases j <;>
simp [complexMatrix, factor, Matrix2.toMatrix, backwardMatrix,
RationalComplex.toComplex, Complex.ext_iff]
private theorem inverse_complex (n : ℕ) :
RationalComplex.toComplex (⟨((n : ℚ)+1)⁻¹,0⟩ : RationalComplex) = ((n : ℂ)+1)⁻¹ := by
have hr (r : ℚ) : RationalComplex.toComplex (⟨r,0⟩ : RationalComplex) = (r : ℂ) := by
apply Complex.ext <;> simp [RationalComplex.toComplex]
simpa using hr (((n : ℚ)+1)⁻¹)
theorem complexMatrix_step (n : ℕ) (s : State) :
complexMatrix (step n s).value =
profileFactor (centerB : ℝ) (centerZ : ℝ) n * complexMatrix s.value := by
simp only [step, complexMatrix_scale, complexMatrix_mul, factor_complex, inverse_complex,
profileFactor, Matrix.smul_mul]
theorem complexMatrix_foldr (ns : List ℕ) :
complexMatrix (ns.foldr step initial).value =
(ns.map (profileFactor (centerB : ℝ) (centerZ : ℝ))).prod := by
induction ns with
| nil => simp [initial]
| cons n ns ih => simp only [List.foldr_cons, complexMatrix_step, ih, List.map_cons, List.prod_cons]
theorem profileFactor_product (b Z : ℝ) (K : ℕ) :
((List.range K).map (profileFactor b Z)).prod = profileProduct b Z K := by
induction K with
| zero => simp [profileProduct, backwardProduct]
| succ K ih =>
rw [List.range_succ, List.map_append, List.prod_append, List.map_singleton,
List.prod_singleton, ih]
simp only [profileProduct, profileFactor, backwardProduct, Matrix.smul_mul,
Matrix.mul_smul, smul_smul]
have he : (-Complex.I * (b : ℂ)) + K = (K : ℂ) - Complex.I*b := by ring
rw [he]
congr 1
simp only [Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, _root_.mul_inv_rev]
ring
theorem result_value_eq_profileProduct :
complexMatrix result.value = profileProduct (centerB : ℝ) (centerZ : ℝ) 34 := by
rw [result, complexMatrix_foldr, profileFactor_product]
end DefocusingNLS.ProfileCertificate
end OAI
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