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