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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileNormalizedBounds
namespace OAI
/-! The zero-tail quotient has the certified linear approximation. -/
namespace DefocusingNLS.ProfileCertificate
private theorem ratio_error (d b l : ℂ) (η χ S : ℝ)
(hη : 0 ≤ η) (hχ : 0 ≤ χ) (hχ1 : χ < 1) (hS : 0 ≤ S)
(hd : ‖d-l‖ ≤ η) (hb : ‖b-1‖ ≤ χ) (hl : ‖l‖ ≤ S) :
‖d/b-l‖ ≤ (η+S*χ)/(1-χ) := by
have hnorm : 1-χ ≤ ‖b‖ := by
have hn := norm_sub_norm_le (1 : ℂ) b
rw [norm_one, norm_sub_rev] at hn
linarith
have hden : 0 < 1-χ := sub_pos.mpr hχ1
have hb0 : b ≠ 0 := norm_pos_iff.mp (hden.trans_le hnorm)
have he : d/b-l = (d-l+l*(1-b))/b := by field_simp [hb0]; ring
rw [he, norm_div]
have hnum : ‖d-l+l*(1-b)‖ ≤ η+S*χ := by
apply (norm_add_le _ _).trans
rw [norm_mul, norm_sub_rev (1 : ℂ) b]
exact add_le_add hd (mul_le_mul hl hb (norm_nonneg _) hS)
exact (div_le_div_of_nonneg_right hnum (norm_nonneg _)).trans
(div_le_div_of_nonneg_left (by positivity) hden hnorm)
theorem normalized_first_denominator_lower (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) :
1-(changeBound : ℝ) ≤ ‖normalizedProfile b z 0 1‖ := by
have h := normalizedProfile_variation b z hb hz 0 1
rw [centralNormalized_zero_one] at h
have hn := norm_sub_norm_le (1 : ℂ) (normalizedProfile b z 0 1)
rw [norm_one, norm_sub_rev] at hn
linarith
theorem normalized_first_entry_ne_zero (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) :
normalizedProfile b z 0 1 ≠ 0 := by
apply norm_pos_iff.mp
have hχ : (changeBound : ℝ) < 1 := by norm_num [changeBound, radius]
linarith [normalized_first_denominator_lower b z hb hz]
theorem normalized_zero_tail_error (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) :
‖normalizedProfile b z 1 1/normalizedProfile b z 0 1-
((b : ℂ)*centralX+(z : ℂ)*centralY)‖ ≤ (jetError : ℝ) := by
have h := ratio_error (normalizedProfile b z 1 1) (normalizedProfile b z 0 1)
((b : ℂ)*centralX+(z : ℂ)*centralY)
((1/1000000000000 : ℝ)+50000*(radius : ℝ)^2) (changeBound : ℝ)
((283/1000 : ℝ)*(radius : ℝ))
(by positivity) (by norm_num [changeBound, radius]) (by norm_num [changeBound, radius])
(by norm_num [radius]) (normalizedProfile_last_remainder b z hb hz).le
(by simpa only [centralNormalized_zero_one] using (normalizedProfile_variation b z hb hz 0 1).le)
(free_linear_size b z hb hz)
convert! h using 1
unfold jetError
push_cast
ring
end DefocusingNLS.ProfileCertificate
end OAI
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