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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileTaylorBound
import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileCentralEnclosures
namespace OAI
/-! Uniform entrywise bounds for the actual K=34 free matching product. -/
open Matrix
namespace DefocusingNLS.ProfileCertificate
attribute [local irreducible] profileProduct
private theorem derivativeB_entry (i j : Fin 2) :
‖complexMatrix result.derivB i j‖ < 20*‖centralM‖ := by
fin_cases i <;> fin_cases j <;> exact central_derivative_entries _ (by simp [Matrix2.toMatrix])
private theorem derivativeZ_entry (i j : Fin 2) :
‖complexMatrix result.derivZ i j‖ < 20*‖centralM‖ := by
fin_cases i <;> fin_cases j <;> exact central_derivative_entries _ (by simp [Matrix2.toMatrix])
private theorem error_entry (i j : Fin 2) :
rationalMatrix result.error i j < 50000*‖centralM‖ := by
fin_cases i <;> fin_cases j <;> exact (central_error_entries _ (by simp [Matrix2.toMatrix])).2
/-- The entrywise complex Taylor remainder in `free:profile-enclosures`. -/
theorem profile_entry_remainder (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) (i j : Fin 2) :
‖profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j-
complexMatrix result.value i j-
(b : ℂ)*complexMatrix result.derivB i j-
(z : ℂ)*complexMatrix result.derivZ i j‖ <
50000*‖centralM‖*(radius : ℝ)^2 := by
have h := profileProduct_remainder b z hb hz i j
have he : 0 < (radius : ℝ)^2 := by norm_num [radius]
have hs := mul_lt_mul_of_pos_left (error_entry i j) he
have hnorm :
‖profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j-
complexMatrix result.value i j-
(b : ℂ)*complexMatrix result.derivB i j-
(z : ℂ)*complexMatrix result.derivZ i j‖ ≤
(radius : ℝ)^2*rationalMatrix result.error i j := by
convert! h using 1
simp only [linearModel, Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply,
Complex.real_smul]
congr 1
ring
exact hnorm.trans_lt (hs.trans_eq (by ring))
/-- The entire matrix varies by less than chi times the central denominator. -/
theorem profile_entry_variation (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) (i j : Fin 2) :
‖profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j-
complexMatrix result.value i j‖ < (changeBound : ℝ)*‖centralM‖ := by
let L := profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j
let U := complexMatrix result.value i j
let X := complexMatrix result.derivB i j
let Y := complexMatrix result.derivZ i j
have hδ : 0 ≤ (radius : ℝ) := by norm_num [radius]
have hx : ‖(b : ℂ)*X‖ ≤ (radius : ℝ)*(20*‖centralM‖) := by
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs]
exact mul_le_mul hb (derivativeB_entry i j).le (norm_nonneg _) hδ
have hy : ‖(z : ℂ)*Y‖ ≤ (radius : ℝ)*(20*‖centralM‖) := by
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs]
exact mul_le_mul hz (derivativeZ_entry i j).le (norm_nonneg _) hδ
calc
‖L-U‖ = ‖(L-U-(b : ℂ)*X-(z : ℂ)*Y)+(b : ℂ)*X+(z : ℂ)*Y‖ := by
congr 1; ring
_ ≤ ‖L-U-(b : ℂ)*X-(z : ℂ)*Y‖+‖(b : ℂ)*X‖+‖(z : ℂ)*Y‖ :=
(norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl)
_ < 50000*‖centralM‖*(radius : ℝ)^2+
(radius : ℝ)*(20*‖centralM‖)+(radius : ℝ)*(20*‖centralM‖) := by
exact add_lt_add_of_lt_of_le
(add_lt_add_of_lt_of_le (profile_entry_remainder b z hb hz i j) hx) hy
_ = _ := by unfold changeBound; push_cast; ring
noncomputable def normalizedProfile (b z : ℝ) (i j : Fin 2) : ℂ :=
profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 i j/centralM
noncomputable def centralNormalized (i j : Fin 2) : ℂ :=
complexMatrix result.value i j/centralM
theorem normalizedProfile_variation (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) (i j : Fin 2) :
‖normalizedProfile b z i j-centralNormalized i j‖ < (changeBound : ℝ) := by
rw [normalizedProfile, centralNormalized, ← sub_div, norm_div]
exact (div_lt_iff₀ (norm_pos_iff.mpr centralM_ne_zero)).mpr
(profile_entry_variation b z hb hz i j)
@[simp] theorem centralNormalized_zero_one : centralNormalized 0 1 = 1 := by
exact div_self centralM_ne_zero
end DefocusingNLS.ProfileCertificate
end OAI
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