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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileEntryBounds
import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileLinearMap
namespace OAI
/-! Normalized numerical bounds before division by the free-profile matching denominator. -/
open Matrix
namespace DefocusingNLS.ProfileCertificate
open RationalComplex
attribute [local irreducible] profileProduct
private theorem lt_norm_of_sq (v : RationalComplex) (c : ℚ) (hc : 0 ≤ c)
(h : c^2 < normSq v) : (c : ℝ) < ‖toComplex v‖ := by
have hh := (Rat.cast_lt (K := ℝ)).mpr h
rw [← normSq_toComplex v, Complex.normSq_eq_norm_sq] at hh
push_cast at hh
have hc' : (0 : ℝ) ≤ c := Rat.cast_nonneg.mpr hc
nlinarith [norm_nonneg (toComplex v)]
private theorem central_u : centralNormalized 0 0 =
toComplex (quotient result.value.a result.value.b) := by
exact (toComplex_quotient _ _).symm
theorem central_u_bound : ‖centralNormalized 0 0‖ < (318/100 : ℝ) := by
rcases linear_bounds with ⟨_, _, _, hu, _⟩
rw [central_u]
simpa only [Rat.cast_div, Rat.cast_ofNat] using
norm_toComplex_lt_of_sq _ (318/100) (by norm_num) hu
theorem central_denominator_bound : (234/100 : ℝ) <
‖1-(Complex.I*(centerZ : ℝ)/5)*centralNormalized 0 0‖ := by
rcases linear_bounds with ⟨_, _, hd, _⟩
have h := lt_norm_of_sq _ (234/100) (by norm_num) hd
have hi : toComplex (⟨0,centerZ/5⟩ : RationalComplex) = Complex.I*(centerZ : ℝ)/5 := by
apply Complex.ext <;> simp [toComplex]
simpa only [map_sub, map_one, map_mul, hi, ← central_u, Rat.cast_div, Rat.cast_ofNat] using h
theorem central_last_entry_bound : ‖centralNormalized 1 1‖ < (1/1000000000000 : ℝ) := by
rcases central_bounds with ⟨_, _, _, _, _, _, hd, _⟩
have h := norm_toComplex_lt_mul result.value.d result.value.b (1/1000000000000)
(by norm_num) hd
change ‖toComplex result.value.d/centralM‖ < _
rw [norm_div]
exact (div_lt_iff₀ (norm_pos_iff.mpr centralM_ne_zero)).mpr (by
simpa only [centralM, Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] using h)
theorem centralX_bound : ‖centralX‖ < (159/1000 : ℝ) := by
rcases linear_bounds with ⟨_, _, _, _, hx, _⟩
simpa only [centralX, Rat.cast_div, Rat.cast_ofNat] using
norm_toComplex_lt_of_sq _ (159/1000) (by norm_num) hx
theorem centralY_bound : ‖centralY‖ < (124/1000 : ℝ) := by
rcases linear_bounds with ⟨_, _, _, _, _, hy, _⟩
simpa only [centralY, Rat.cast_div, Rat.cast_ofNat] using
norm_toComplex_lt_of_sq _ (124/1000) (by norm_num) hy
theorem free_linear_size (b z : ℝ) (hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) :
‖(b : ℂ)*centralX+(z : ℂ)*centralY‖ ≤ (283/1000 : ℝ)*(radius : ℝ) := by
have hδ : 0 ≤ (radius : ℝ) := by norm_num [radius]
calc
_ ≤ ‖(b : ℂ)*centralX‖+‖(z : ℂ)*centralY‖ := norm_add_le _ _
_ = |b| *‖centralX‖+|z| *‖centralY‖ := by simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs]
_ ≤ (radius : ℝ)*(159/1000)+(radius : ℝ)*(124/1000) :=
add_le_add (mul_le_mul hb centralX_bound.le (norm_nonneg _) hδ)
(mul_le_mul hz centralY_bound.le (norm_nonneg _) hδ)
_ = _ := by ring
theorem normalizedProfile_last_remainder (b z : ℝ)
(hb : |b| ≤ (radius : ℝ)) (hz : |z| ≤ (radius : ℝ)) :
‖normalizedProfile b z 1 1-((b : ℂ)*centralX+(z : ℂ)*centralY)‖ <
(1/1000000000000 : ℝ)+50000*(radius : ℝ)^2 := by
have hx : toComplex result.derivB.d/centralM = centralX := (toComplex_quotient _ _).symm
have hy : toComplex result.derivZ.d/centralM = centralY := (toComplex_quotient _ _).symm
have he : normalizedProfile b z 1 1-centralNormalized 1 1-
((b : ℂ)*centralX+(z : ℂ)*centralY) =
(profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 1 1-
toComplex result.value.d-(b : ℂ)*toComplex result.derivB.d-
(z : ℂ)*toComplex result.derivZ.d)/centralM := by
rw [← hx, ← hy]
change profileProduct ((centerB : ℝ)+b) ((centerZ : ℝ)+z) 34 1 1/centralM-
toComplex result.value.d/centralM-
((b : ℂ)*(toComplex result.derivB.d/centralM)+(z : ℂ)*(toComplex result.derivZ.d/centralM)) = _
ring
have hr : ‖normalizedProfile b z 1 1-centralNormalized 1 1-
((b : ℂ)*centralX+(z : ℂ)*centralY)‖ < 50000*(radius : ℝ)^2 := by
rw [he, norm_div]
apply (div_lt_iff₀ (norm_pos_iff.mpr centralM_ne_zero)).mpr
convert! profile_entry_remainder b z hb hz 1 1 using 1
ring
calc
_ = ‖(normalizedProfile b z 1 1-centralNormalized 1 1-
((b : ℂ)*centralX+(z : ℂ)*centralY))+centralNormalized 1 1‖ := by congr 1; ring
_ ≤ ‖normalizedProfile b z 1 1-centralNormalized 1 1-
((b : ℂ)*centralX+(z : ℂ)*centralY)‖+‖centralNormalized 1 1‖ := norm_add_le _ _
_ < _ := by linarith [central_last_entry_bound]
end DefocusingNLS.ProfileCertificate
end OAI
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