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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileMatchingQuotient
import OAI.MathematicalPhysics.DefocusingNLS.Profile.FreeSlowContinuity
namespace OAI
/-! Continuity of the actual free-profile quotient on the certified disk. -/
namespace DefocusingNLS.ProfileCertificate
noncomputable def diskProfile (w : Metric.closedBall (0 : ℂ) (radius : ℝ)) : ℂ :=
freeProfileJ ((centerB : ℝ)+w.val.re) ((centerZ : ℝ)+w.val.im)
theorem disk_coordinates (w : Metric.closedBall (0 : ℂ) (radius : ℝ)) :
|w.val.re| ≤ (radius : ℝ) ∧ |w.val.im| ≤ (radius : ℝ) := by
have h : ‖w.val‖ ≤ (radius : ℝ) := by simpa only [Metric.mem_closedBall, dist_zero_right] using w.property
exact ⟨(Complex.abs_re_le_norm _).trans h, (Complex.abs_im_le_norm _).trans h⟩
theorem continuous_diskProfile : Continuous diskProfile := by
let p : Metric.closedBall (0 : ℂ) (radius : ℝ) → ℝ × ℝ :=
fun w => ((centerB : ℝ)+w.val.re, (centerZ : ℝ)+w.val.im)
have hp : Continuous p := by unfold p; fun_prop
have hm : ∀ w, p w ∈ freeParameterRectangle := by
intro w
obtain ⟨hb, hz⟩ := disk_coordinates w
dsimp [p, freeParameterRectangle]
have hc : 0 ≤ (centerB : ℝ) := by norm_num [centerB]
have hb' := (abs_add_le (centerB : ℝ) w.val.re).trans
(add_le_add (le_of_eq (abs_of_nonneg hc)) hb)
obtain ⟨hzl, hzu⟩ := abs_le.mp hz
norm_num [centerB, centerZ, radius] at hb hb' hzl hzu ⊢
exact ⟨by linarith, by linarith, by linarith⟩
have h0 := (continuousOn_free_regularizedSlowSolution 0 (by norm_num) 6).comp_continuous hp hm
have h1 := (continuousOn_free_regularizedSlowSolution 1 (by norm_num) 7).comp_continuous hp hm
dsimp only [Function.comp_def] at h0 h1
have hq : Continuous (fun w => freeShiftedQ 0 (p w)) := by
unfold freeShiftedQ p
fun_prop
have he (w) : freeShiftedQ 1 (p w) = freeShiftedQ 0 (p w)+1 := by
unfold freeShiftedQ
push_cast
ring
simp_rw [he] at h1
have hn (w) : regularizedSlowSolution (freeShiftedQ 0 (p w)) 6
(freeSpatialArgument (p w)) ≠ 0 := by
obtain ⟨hb, hz⟩ := disk_coordinates w
simpa only [freeShiftedQ, freeSpatialArgument, p, Complex.ofReal_zero, zero_sub, neg_mul] using
free_profile_nonzero w.val.re w.val.im hb hz
convert! (hq.mul h1).div h0 hn using 1
unfold diskProfile freeProfileJ freeShiftedQ freeSpatialArgument p
funext w
simp only [Pi.mul_apply, Pi.div_apply, Complex.ofReal_zero, zero_sub, neg_mul]
end DefocusingNLS.ProfileCertificate
end OAI
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