openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileProductMatching.lean
| 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 | |