File size: 3,360 Bytes
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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileExistence
import Mathlib.Topology.UniformSpace.UniformConvergence
namespace OAI
/-! The four-real-dimensional degree step for simultaneous value and slope matching. -/
open Filter
namespace DefocusingNLS.ProfileCertificate
abbrev ProfileMatchingBall := Metric.closedBall (0 : ℂ × ℂ) (radius : ℝ)
noncomputable def profileMatchingParameter (z : ProfileMatchingBall) :
Metric.closedBall (0 : ℂ) (radius : ℝ) :=
⟨z.val.2,by
have hz : ‖z.val‖ ≤ (radius : ℝ) := by
simpa only [Metric.mem_closedBall,dist_zero_right] using z.property
simpa only [Metric.mem_closedBall,dist_zero_right] using (norm_snd_le z.val).trans hz⟩
noncomputable def profileProductLinearEquiv : (ℂ × ℂ) ≃L[ℝ] (ℂ × ℂ) :=
(ContinuousLinearEquiv.refl ℝ ℂ).prodCongr centralLinearEquiv
theorem profileProductLinearEquiv_inverse_bound (z : ℂ × ℂ) :
‖profileProductLinearEquiv.symm z‖ ≤ ‖z‖/(48/1000 : ℝ) := by
change max ‖z.1‖ ‖centralLinearEquiv.symm z.2‖ ≤ _
apply max_le
· apply (norm_fst_le z).trans
rw [le_div_iff₀ (by norm_num)]
nlinarith [norm_nonneg z]
· exact (centralLinearEquiv_inverse_bound z.2).trans
((div_le_div_iff_of_pos_right (by norm_num)).mpr (norm_snd_le z))
theorem zero_of_profile_product_error (f : ProfileMatchingBall → ℂ × ℂ)
(hf : Continuous f)
(he : ∀ z, ‖f z-profileProductLinearEquiv z.val‖ ≤ (50/1000000000000 : ℝ)) :
∃ z, f z=0 := by
apply zero_of_near_invertible_linear_map profileProductLinearEquiv (radius : ℝ)
((50/1000000000000)/(48/1000)) (by norm_num [radius]) (by norm_num [radius]) f hf
intro z
have hid : profileProductLinearEquiv.symm (f z)-z.val=
profileProductLinearEquiv.symm (f z-profileProductLinearEquiv z.val) := by
rw [map_sub,profileProductLinearEquiv.symm_apply_apply]
rw [hid]
exact (profileProductLinearEquiv_inverse_bound _).trans
((div_le_div_iff_of_pos_right (by norm_num)).mpr (he z))
theorem zero_of_profile_product_limit (f : ℕ → ProfileMatchingBall → ℂ × ℂ)
(hf : ∀ᶠ n in atTop, Continuous (f n))
(hlim : TendstoUniformly f (fun z => (z.val.1,diskProfile (profileMatchingParameter z))) atTop) :
∀ᶠ n in atTop, ∃ z, f n z=0 := by
have hclose := (Metric.tendstoUniformly_iff.mp hlim)
(25/1000000000000) (by norm_num)
filter_upwards [hf,hclose] with n hn he
apply zero_of_profile_product_error (f n) hn
intro z
have herr : ‖(z.val.1,diskProfile (profileMatchingParameter z))-
profileProductLinearEquiv z.val‖ ≤ (25/1000000000000 : ℝ) := by
change max ‖z.val.1-z.val.1‖
‖diskProfile (profileMatchingParameter z)-centralLinearEquiv z.val.2‖ ≤ _
rw [sub_self,norm_zero,max_eq_right (norm_nonneg _)]
exact (diskProfile_linear_error (profileMatchingParameter z)).le
have hsmall : ‖f n z-(z.val.1,diskProfile (profileMatchingParameter z))‖ <
(25/1000000000000 : ℝ) := by
simpa only [dist_comm,dist_eq_norm] using he z
have htri := norm_add_le
(f n z-(z.val.1,diskProfile (profileMatchingParameter z)))
((z.val.1,diskProfile (profileMatchingParameter z))-profileProductLinearEquiv z.val)
rw [sub_add_sub_cancel] at htri
linarith
end DefocusingNLS.ProfileCertificate
end OAI
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