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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 | 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
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