openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileZeroCriterion.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileLinearMap | |
| namespace OAI | |
| /-! Brouwer's theorem applied to the specific certified central linear map. -/ | |
| namespace DefocusingNLS.ProfileCertificate | |
| theorem central_linear_injective : Function.Injective (realLinearMap centralX centralY) := by | |
| intro x y hxy | |
| have hz : realLinearMap centralX centralY (x-y) = 0 := by rw [map_sub, hxy, sub_self] | |
| have hb := central_minimum_stretch (x-y) | |
| rw [hz, norm_zero] at hb | |
| have hn : ‖x-y‖ = 0 := by nlinarith [norm_nonneg (x-y)] | |
| exact sub_eq_zero.mp (norm_eq_zero.mp hn) | |
| noncomputable def centralLinearEquiv : ℂ ≃L[ℝ] ℂ := | |
| (LinearEquiv.ofBijective (realLinearMap centralX centralY).toLinearMap | |
| ⟨central_linear_injective, LinearMap.injective_iff_surjective.mp central_linear_injective⟩).toContinuousLinearEquiv | |
| @[simp] theorem centralLinearEquiv_apply (z : ℂ) : | |
| centralLinearEquiv z = realLinearMap centralX centralY z := rfl | |
| theorem centralLinearEquiv_inverse_bound (z : ℂ) : | |
| ‖centralLinearEquiv.symm z‖ ≤ ‖z‖/(48/1000 : ℝ) := by | |
| have h := central_minimum_stretch (centralLinearEquiv.symm z) | |
| change (48/1000 : ℝ)*‖centralLinearEquiv.symm z‖ ≤ ‖centralLinearEquiv (centralLinearEquiv.symm z)‖ at h | |
| rw [centralLinearEquiv.apply_symm_apply] at h | |
| exact (le_div_iff₀ (by norm_num)).mpr (by nlinarith) | |
| /-- The precise error margin needed to force a zero on the free-profile disk. | |
| Continuity and the approximation are hypotheses on a finite-dimensional map, | |
| not assumed statements about the NLS or its profiles. -/ | |
| theorem zero_of_free_profile_error | |
| (f : Metric.closedBall (0 : ℂ) (radius : ℝ) → ℂ) (hf : Continuous f) | |
| (herror : ∀ z, ‖f z-realLinearMap centralX centralY z.val‖ ≤ 25/1000000000000) : | |
| ∃ z, f z = 0 := by | |
| apply zero_of_near_invertible_linear_map centralLinearEquiv (radius : ℝ) | |
| ((25/1000000000000)/(48/1000)) (by norm_num [radius]) (by norm_num [radius]) f hf | |
| intro z | |
| have he : centralLinearEquiv.symm (f z)-z.val = | |
| centralLinearEquiv.symm (f z-centralLinearEquiv z.val) := by | |
| rw [map_sub, centralLinearEquiv.symm_apply_apply] | |
| rw [he] | |
| exact (centralLinearEquiv_inverse_bound _).trans | |
| ((div_le_div_iff_of_pos_right (by norm_num)).mpr (herror z)) | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |