| import Mathlib |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.MovingHessianLimit |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.BilinearCompact |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.BilinearMatrix |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.BilinearClose |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter |
| open scoped Topology |
|
|
| namespace WeakMTWTransport |
| section CenterDetCompact |
| variable {E:Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
|
|
| local instance centerDetDualGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance |
| local instance centerDetDualSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance |
| local instance centerDetBilinearGroup : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance |
| local instance centerDetBilinearSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance |
|
|
| lemma exists_center_matrix_limit {V C:ℕ → E →L[ℝ] E →L[ℝ] ℝ} |
| {C₀:E →L[ℝ] E →L[ℝ] ℝ} {a:ℕ → ℝ} {a₀ k₀:ℝ} |
| (hC:Tendsto C atTop (𝓝 C₀)) (ha:Tendsto a atTop (𝓝 a₀)) (hk:0<k₀) |
| (hV:∀ᶠ j in atTop,(∀d e,V j d e=V j e d) ∧ |
| (∀d,0 ≤ V j d d) ∧ (∀d,V j d d ≤ C j d d) ∧ |
| 0 ≤ a j ∧ k₀ ≤ a j*(bilinearOperator (V j)).det) : |
| ∃V₀:E →L[ℝ] E →L[ℝ] ℝ,∃σ:ℕ → ℕ,StrictMono σ ∧ |
| Tendsto (V ∘ σ) atTop (𝓝 V₀) ∧ (∀d e,V₀ d e=V₀ e d) ∧ |
| (∀d,d≠0 → 0<V₀ d d) ∧ (∀d,V₀ d d ≤ C₀ d d) ∧ |
| 0<a₀ ∧ k₀ ≤ a₀*(bilinearOperator V₀).det := by |
| have hCn : Tendsto (fun j => ‖C j‖) atTop (𝓝 ‖C₀‖) := hC.norm |
| have hCb:∀ᶠ j in atTop,‖C j‖≤‖C₀‖+1:=by |
| filter_upwards [hCn.eventually (gt_mem_nhds (show ‖C₀‖<‖C₀‖+1 by linarith))] with j hj |
| exact hj.le |
| have hbound:∀ᶠ j in atTop,‖V j‖≤‖C₀‖+1:=by |
| filter_upwards [hCb,hV] with j hj hv |
| apply symmetric_bilinear_norm_bound hv.1 (by positivity) |
| intro d |
| rw [abs_of_nonneg (hv.2.1 d)] |
| calc |
| V j d d ≤ C j d d:=hv.2.2.1 d |
| _ ≤ |C j d d|:=le_abs_self _ |
| _ ≤ ‖C j‖*‖d‖^2:=bilinear_quadratic_norm_bound _ _ |
| _ ≤ (‖C₀‖+1)*‖d‖^2:=mul_le_mul_of_nonneg_right hj (sq_nonneg _) |
| obtain ⟨V₀,σ,hσ,HV⟩:=exists_bilinear_limit hbound |
| have hve:=hσ.tendsto_atTop.eventually hV |
| have hvs:∀d e,V₀ d e=V₀ e d:=bilinear_limit_symm HV (hve.mono (fun j hj=>hj.1)) |
| have hvp:∀d,0≤V₀ d d:=bilinear_limit_nonneg HV (hve.mono (fun j hj=>hj.2.1)) |
| have hae:0≤a₀:=ge_of_tendsto (ha.comp hσ.tendsto_atTop) (hve.mono (fun j hj=>hj.2.2.2.1)) |
| have hdet:k₀≤a₀*(bilinearOperator V₀).det:=ge_of_tendsto |
| ((ha.comp hσ.tendsto_atTop).mul (bilinear_det_tendsto HV)) (hve.mono (fun j hj=>hj.2.2.2.2)) |
| have hdp:0 < (bilinearOperator V₀).det:=by |
| have H:=bilinear_det_nonneg hvs hvp |
| nlinarith only [hk,hdet,hae,H] |
| have hap:0<a₀:=by nlinarith only [hk,hdet,hae,hdp] |
| refine ⟨V₀,σ,hσ,HV,hvs,bilinear_positive_of_det_ne_zero hvs hvp hdp.ne',?_,hap,hdet⟩ |
| intro d |
| exact le_of_tendsto_of_tendsto |
| (tendsto_clm_apply (tendsto_clm_apply HV tendsto_const_nhds) tendsto_const_nhds) |
| (tendsto_clm_apply (tendsto_clm_apply (hC.comp hσ.tendsto_atTop) tendsto_const_nhds) tendsto_const_nhds) |
| (hve.mono (fun j hj=>hj.2.2.1 d)) |
| |
| lemma exists_center_matrix_limit_variable {V C:ℕ → E →L[ℝ] E →L[ℝ] ℝ} |
| {C₀:E →L[ℝ] E →L[ℝ] ℝ} {a k:ℕ → ℝ} {a₀ k₀:ℝ} |
| (hC:Tendsto C atTop (𝓝 C₀)) (ha:Tendsto a atTop (𝓝 a₀)) |
| (hk:Tendsto k atTop (𝓝 k₀)) (hk0:0<k₀) |
| (hV:∀ᶠ j in atTop,(∀d e,V j d e=V j e d) ∧ |
| (∀d,0 ≤ V j d d) ∧ (∀d,V j d d ≤ C j d d) ∧ |
| 0 ≤ a j ∧ k j ≤ a j*(bilinearOperator (V j)).det) : |
| ∃V₀:E →L[ℝ] E →L[ℝ] ℝ,∃σ:ℕ → ℕ,StrictMono σ ∧ |
| Tendsto (V ∘ σ) atTop (𝓝 V₀) ∧ (∀d e,V₀ d e=V₀ e d) ∧ |
| (∀d,d≠0 → 0<V₀ d d) ∧ (∀d,V₀ d d ≤ C₀ d d) ∧ |
| 0<a₀ ∧ k₀ ≤ a₀*(bilinearOperator V₀).det := by |
| have HV:∀ᶠ j in atTop,(∀d e,V j d e=V j e d) ∧ |
| (∀d,0 ≤ V j d d) ∧ (∀d,V j d d ≤ C j d d) ∧ |
| 0 ≤ a j ∧ k₀/2 ≤ a j*(bilinearOperator (V j)).det:=by |
| filter_upwards [hV,hk.eventually (lt_mem_nhds (show k₀/2<k₀ by linarith))] with j hj hkj |
| exact ⟨hj.1,hj.2.1,hj.2.2.1,hj.2.2.2.1,hkj.le.trans hj.2.2.2.2⟩ |
| obtain ⟨V₀,σ,hσ,HV,hVs,hVp,hVC,ha0,_⟩:= |
| exists_center_matrix_limit hC ha (half_pos hk0) HV |
| refine ⟨V₀,σ,hσ,HV,hVs,hVp,hVC,ha0,?_⟩ |
| exact le_of_tendsto_of_tendsto (hk.comp hσ.tendsto_atTop) |
| ((ha.comp hσ.tendsto_atTop).mul (bilinear_det_tendsto HV)) |
| ((hσ.tendsto_atTop.eventually hV).mono (fun j hj=>hj.2.2.2.2)) |
| end CenterDetCompact |
| end WeakMTWTransport |
| |
| end |
| end |
| |
| end OAI |
| |
