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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileTailError
import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileQuotientContinuity
import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileZeroCriterion
namespace OAI
/-! A zero of the actual free-profile matching quotient in the certified disk. -/
namespace DefocusingNLS.ProfileCertificate
theorem diskProfile_linear_error (w : Metric.closedBall (0 : ℂ) (radius : ℝ)) :
‖diskProfile w-realLinearMap centralX centralY w.val‖ < (25/1000000000000 : ℝ) := by
obtain ⟨hb, hz⟩ := disk_coordinates w
simpa only [diskProfile, realLinearMap_apply] using
free_profile_linear_error w.val.re w.val.im hb hz
theorem diskProfile_bound (w : Metric.closedBall (0 : ℂ) (radius : ℝ)) :
‖diskProfile w‖ < (3/1000000000 : ℝ) := by
obtain ⟨hb, hz⟩ := disk_coordinates w
have he := diskProfile_linear_error w
have hl := free_linear_size w.val.re w.val.im hb hz
rw [← realLinearMap_apply] at hl
have hc := (Rat.cast_lt (K := ℝ)).mpr propagation_comparisons.2.2.2.1
push_cast at hc
have hn : ‖diskProfile w‖ ≤
‖diskProfile w-realLinearMap centralX centralY w.val‖+
‖realLinearMap centralX centralY w.val‖ := by
calc
_ = ‖(diskProfile w-realLinearMap centralX centralY w.val)+
realLinearMap centralX centralY w.val‖ := by rw [sub_add_cancel]
_ ≤ _ := norm_add_le _ _
linarith
theorem diskProfile_boundary_margin (w : Metric.closedBall (0 : ℂ) (radius : ℝ))
(hw : ‖w.val‖ = (radius : ℝ)) :
(91/200000000000 : ℝ) < ‖diskProfile w‖ := by
have hl := central_minimum_stretch w.val
rw [hw] at hl
have he := diskProfile_linear_error w
have hn := norm_sub_norm_le (realLinearMap centralX centralY w.val) (diskProfile w)
rw [norm_sub_rev] at hn
have hc : (48/1000 : ℝ)*(radius : ℝ) = (48/100000000000 : ℝ) := by norm_num [radius]
rw [hc] at hl
linarith
/-- The certified error and Brouwer's theorem give an actual free matching root. -/
theorem exists_diskProfile_zero :
∃ w : Metric.closedBall (0 : ℂ) (radius : ℝ),
‖w.val‖ < (radius : ℝ) ∧ diskProfile w = 0 := by
obtain ⟨w, hw⟩ := zero_of_free_profile_error diskProfile continuous_diskProfile
(fun w => (diskProfile_linear_error w).le)
refine ⟨w, ?_, hw⟩
have hn : ‖w.val‖ ≤ (radius : ℝ) := by
simpa only [Metric.mem_closedBall, dist_zero_right] using w.property
apply lt_of_le_of_ne hn
intro he
have hm := diskProfile_boundary_margin w he
rw [hw, norm_zero] at hm
norm_num at hm
/-- The free matching condition is the vanishing of the actual spatial derivative. -/
theorem freeProfileJ_eq_log_derivative (b Z : ℝ) (hZ : 0 < Z) :
freeProfileJ b Z =
-deriv (regularizedSlowSolution (-Complex.I*(b : ℂ)) 6) (-Complex.I*(Z : ℂ))/
regularizedSlowSolution (-Complex.I*(b : ℂ)) 6 (-Complex.I*(Z : ℂ)) := by
have h := (hasDerivAt_regularizedSlowSolution_shift_closed
(-Complex.I*(b : ℂ)) 6 (-Complex.I*(Z : ℂ))
(by simp) (by simp) (by simpa using hZ.ne')).deriv
rw [h]
unfold freeProfileJ
ring
theorem exists_free_profile_matching :
∃ b Z : ℝ, (b-(centerB : ℝ))^2+(Z-(centerZ : ℝ))^2 < (radius : ℝ)^2 ∧
regularizedSlowSolution (-Complex.I*(b : ℂ)) 6 (-Complex.I*(Z : ℂ)) ≠ 0 ∧
deriv (regularizedSlowSolution (-Complex.I*(b : ℂ)) 6) (-Complex.I*(Z : ℂ)) = 0 := by
obtain ⟨w, hwn, hw⟩ := exists_diskProfile_zero
obtain ⟨hb, hz⟩ := disk_coordinates w
refine ⟨(centerB : ℝ)+w.val.re, (centerZ : ℝ)+w.val.im, ?_, ?_, ?_⟩
· have hsq := (sq_lt_sq₀ (norm_nonneg w.val) (by norm_num [radius])).mpr hwn
rw [← Complex.normSq_eq_norm_sq, Complex.normSq_apply] at hsq
simpa only [add_sub_cancel_left, pow_two] using hsq
· simpa only [neg_mul] using free_profile_nonzero w.val.re w.val.im hb hz
· have hZ : 0 < (centerZ : ℝ)+w.val.im := by
have hz' := (abs_le.mp hz).1
norm_num [centerZ, radius] at hz' ⊢
linarith
change freeProfileJ ((centerB : ℝ)+w.val.re) ((centerZ : ℝ)+w.val.im) = 0 at hw
rw [freeProfileJ_eq_log_derivative _ _ hZ] at hw
have hn := free_profile_nonzero w.val.re w.val.im hb hz
simp only [neg_mul] at hn
simp only [neg_mul] at hw ⊢
exact neg_eq_zero.mp ((div_eq_zero_iff.mp hw).resolve_right hn)
end DefocusingNLS.ProfileCertificate
end OAI
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