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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 96 97 98 99 100 101 102 103 104 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileDeterminantBound
import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileJetError
import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileMatchingQuotient
namespace OAI
/-! Propagation of the strict tail disk to the actual free-profile quotient. -/
open Matrix
namespace DefocusingNLS.ProfileCertificate
attribute [local irreducible] profileProduct normalizedMatchingB normalizedMatchingC
theorem normalized_profile_determinant_bound (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) :
‖Matrix.det (normalizedProfile b z)‖ < (62/10000000000000 : ℝ) := by
have hm : normalizedProfile b z =
centralM⁻¹ • profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 := by
ext i j
simp only [normalizedProfile, Matrix.smul_apply, smul_eq_mul, div_eq_mul_inv]
ring
rw [hm, Matrix.det_smul, Fintype.card_fin, norm_mul, norm_pow, norm_inv]
have h := actual_profile_determinant_bound b z hb
change ‖(profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34).det‖ <
(62/10000000000000 : ℝ)*‖centralM‖^2 at h
have hn : 0 < ‖centralM‖^2 := pow_pos (norm_pos_iff.mpr centralM_ne_zero) _
have hd := (div_lt_iff₀ hn).mpr h
convert! hd using 1
simp only [div_eq_mul_inv, inv_pow]
ring
theorem actual_tail_error (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) :
‖freeProfileJ ((centerB : ℝ)+b) ((centerZ : ℝ)+z)-
normalizedProfile b z 1 1/normalizedProfile b z 0 1‖ < (tailError : ℝ) := by
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 hδ : 0 ≤ (radius : ℝ) := by norm_num [radius]
have hZ : 0 < (centerZ : ℝ)+z := by
have hz' := (abs_le.mp hz).1
norm_num [centerZ, radius] at hz' ⊢
linarith
have hsn : ‖s/5‖ = ((centerZ : ℝ)+z)/5 := by
simp only [s, norm_div, norm_mul, Complex.norm_I, one_mul, Complex.norm_real,
Real.norm_eq_abs, Complex.norm_ofNat, abs_of_pos hZ]
have hr : ‖r+s/5‖ ≤ ((centerZ : ℝ)+z)/5 := (profile_tail_disk b z hz).2
have hru : ‖r‖ ≤ 2*((centerZ : ℝ)+(radius : ℝ))/5 := by
have ht : ‖r‖ ≤ ‖r+s/5‖+‖s/5‖ := by
calc
_ = ‖(r+s/5)-s/5‖ := by rw [add_sub_cancel_right]
_ ≤ _ := norm_sub_le _ _
rw [hsn] at ht
linarith [(abs_le.mp hz).2]
have hgap := normalized_denominator_gap b z hb hz
simp only [← Complex.ofReal_add] at hgap
have hden := matched_denominator_lower (normalizedProfile b z) r (s/5)
(((centerZ : ℝ)+z)/5) hr
have hd : (6/10 : ℝ) ≤ ‖normalizedProfile b z 0 0*r+normalizedProfile b z 0 1‖ := by
dsimp only [s] at hden
linarith
have hχ : 0 < 1-(changeBound : ℝ) := by norm_num [changeBound, radius]
have ht := matched_quotient_bound (normalizedProfile b z) r
(1-(changeBound : ℝ)) (6/10) hχ (by norm_num)
(normalized_first_denominator_lower b z hb hz) hd
rw [freeProfileJ_eq_matched b z hz]
change ‖(normalizedProfile b z 1 0*r+normalizedProfile b z 1 1)/
(normalizedProfile b z 0 0*r+normalizedProfile b z 0 1)-
normalizedProfile b z 1 1/normalizedProfile b z 0 1‖ < _
apply ht.trans_lt
have hdet := normalized_profile_determinant_bound b z hb
have hu : 0 < 2*((centerZ : ℝ)+(radius : ℝ))/5 := by norm_num [centerZ, radius]
have hm : ‖r‖*‖Matrix.det (normalizedProfile b z)‖ <
(2*((centerZ : ℝ)+(radius : ℝ))/5)*(62/10000000000000) :=
(mul_le_mul_of_nonneg_right hru (norm_nonneg _)).trans_lt
(mul_lt_mul_of_pos_left hdet hu)
have hquot := (div_lt_div_iff_of_pos_right (mul_pos hχ (by norm_num : (0 : ℝ)<6/10))).mpr hm
convert! hquot using 1
unfold tailError
push_cast
ring
/-- The certified small error applies to the genuine logarithmic derivative. -/
theorem free_profile_linear_error (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) :
‖freeProfileJ ((centerB : ℝ)+b) ((centerZ : ℝ)+z)-
((b : ℂ)*centralX+(z : ℂ)*centralY)‖ < (25/1000000000000 : ℝ) := by
have ht := actual_tail_error b z hb hz
have hj := normalized_zero_tail_error b z hb hz
have hsum := (Rat.cast_lt (K := ℝ)).mpr propagation_comparisons.2.2.1
push_cast at hsum
calc
_ = ‖(freeProfileJ ((centerB : ℝ)+b) ((centerZ : ℝ)+z)-
normalizedProfile b z 1 1/normalizedProfile b z 0 1)+
(normalizedProfile b z 1 1/normalizedProfile b z 0 1-
((b : ℂ)*centralX+(z : ℂ)*centralY))‖ := by congr 1; ring
_ ≤ _ := norm_add_le _ _
_ < (tailError : ℝ)+(jetError : ℝ) := add_lt_add_of_lt_of_le ht hj
_ < _ := by linarith
end DefocusingNLS.ProfileCertificate
end OAI
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