| import Mathlib |
| import OAI.Analysis.BiholderTransport.Regularity.TemplateRightWeight |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set |
| open scoped BigOperators |
|
|
| namespace WeakMTWTransport |
| section ActiveDifference |
| variable {ι E : Type*} [Fintype ι] [DecidableEq ι] [AddCommGroup E] [Module ℝ E] |
|
|
| omit [DecidableEq ι] in |
| lemma active_difference_ne_bot {pj : ι → E} {p : E} {m : ι → ℝ} |
| {level : E → ℝ} {Bc Bo : ℝ → ℝ} (hsum : ∑ i,m i=1) |
| (hprofiles : Bc (level p) ≤ Bo (level p)) |
| (hparameter : 0 < Bc (level p)-∑ i,m i*Bo (level (pj i))) : |
| Submodule.span ℝ (range (fun i => pj i-p))≠⊥ := by |
| intro H |
| have hpj : ∀ i,pj i=p := by |
| intro i |
| have HH := (Submodule.subset_span (mem_range_self i) : |
| pj i-p∈Submodule.span ℝ (range (fun i => pj i-p))) |
| rw [H] at HH |
| exact sub_eq_zero.mp HH |
| simp_rw [hpj] at hparameter |
| rw [←Finset.sum_mul,hsum,one_mul] at hparameter |
| linarith |
|
|
| omit [DecidableEq ι] in |
| lemma line_active_displacements {pj : ι → E} {p e : E} {m : ι → ℝ} |
| (hline : Submodule.span ℝ (range (fun i => pj i-p))=Submodule.span ℝ {e}) |
| (he : e≠0) (hsum : ∑ i,m i=1) (hbar : ∑ i,m i • pj i=p) : |
| ∃ d : ι → ℝ, (∀ i,pj i=p+d i • e) ∧ ∑ i,m i*d i=0 := by |
| have H : ∀ i,∃ r:ℝ,r • e=pj i-p := by |
| intro i |
| apply Submodule.mem_span_singleton.mp |
| rw [←hline] |
| exact Submodule.subset_span (mem_range_self i) |
| choose d hd using H |
| have hd' : ∀ i,pj i=p+d i • e := by |
| intro i |
| rw [hd i,add_sub_cancel] |
| refine ⟨d,hd',?_⟩ |
| have Hbar : (∑ i,m i*d i) • e=0 := by |
| rw [Finset.sum_smul] |
| simp_rw [mul_smul,hd,smul_sub] |
| rw [Finset.sum_sub_distrib,←Finset.sum_smul,hsum,one_smul,hbar,sub_self] |
| exact (smul_eq_zero.mp Hbar).resolve_right he |
|
|
| end ActiveDifference |
| end WeakMTWTransport |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| open scoped BigOperators |
|
|
| namespace WeakMTWTransport |
| section OutwardGeometry |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| lemma norm_increment_unit_line {p e : E} (he : ‖e‖=1) (d : ℝ) : |
| ‖p+d • e‖^2-‖p‖^2=2*(inner ℝ p e)*d+d^2 := by |
| have hee : inner ℝ e e=1 := by rw [real_inner_self_eq_norm_sq,he]; norm_num |
| rw [←real_inner_self_eq_norm_sq] |
| simp only [inner_add_left,inner_add_right,real_inner_smul_left,inner_smul_right, |
| real_inner_comm e p,hee,real_inner_self_eq_norm_sq] |
| ring |
|
|
| lemma outward_active_endpoint {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {n : ℕ} (hn : Fintype.card ι ≤ n+1) {pj : ι → E} {p e : E} |
| {m B si : ι → ℝ} {D c eta delta zeta s center a0 : ℝ} |
| (hm : ∀ i,0 < m i) (hsum : ∑ i,m i=1) (hbar : ∑ i,m i • pj i=p) |
| (he : ‖e‖=1) (hq : 0 < inner ℝ p e) |
| (hline : Submodule.span ℝ (range (fun i => pj i-p))=Submodule.span ℝ {e}) |
| (hD : 0 < D) (hc : 0 < c) (ha0 : 0 < a0) (hqa : a0*D ≤ (inner ℝ p e)^2) |
| (heta : eta ≤ 1/128) (hdelta : delta ≤ zeta) (hzeta : zeta ≤ 1/16) |
| (hs : s < 3/4) (hlow : ∀ i,-2*eta ≤ si i) |
| (hrange : ∀ i,|si i-s| ≤ 1+delta) |
| (hnorm : ∀ i,|‖pj i‖^2-‖p‖^2+2*D*(si i-s)| ≤ zeta*D) |
| (hB : ∀ i,0 ≤ B i) (hleft : ∀ i,si i < 1-eta → 1 ≤ B i) |
| (hcenter : center ≤ 1) (hparameter : c ≤ center-∑ i,m i*B i) : |
| let mstar := c/(12*(n+1:ℝ)) |
| let theta0 := min (mstar/2) (a0/3) |
| ∃ i d, 0 < d ∧ pj i=p+d • e ∧ mstar ≤ m i ∧ |
| c*D/(8*(n+1:ℝ)*(inner ℝ p e)) ≤ m i*d ∧ |
| d ≤ 3*D/(2*(inner ℝ p e)) ∧ -2*eta ≤ si i ∧ si i < 1-eta ∧ |
| B i ≤ 12*(n+1:ℝ)/c ∧ 0 < theta0 ∧ |
| ∀ theta∈Icc (-theta0) 1, |
| p+theta • (pj i-p)∈convexHull ℝ (range pj) ∧ |
| (inner ℝ p e)/2 ≤ inner ℝ (p+theta • (pj i-p)) e ∧ |
| ‖pj i-p‖^2/(inner ℝ (p+theta • (pj i-p)) (pj i-p)) ≤ 3/a0 := by |
| intro mstar theta0 |
| have he0 : e≠0 := by intro H; rw [H,norm_zero] at he; norm_num at he |
| obtain ⟨d,hd,hbard⟩ := line_active_displacements hline he0 hsum hbar |
| have hnorm' : ∀ i,|2*(inner ℝ p e)*d i+d i^2+2*D*(si i-s)| ≤ zeta*D := by |
| intro i |
| have H := hnorm i |
| rwa [hd i,norm_increment_unit_line he (d i)] at H |
| obtain ⟨i,hdi,hmi,hprod,hdub,hli,hri,hBi⟩ := outward_from_levels hn hm hsum hbard hq hD hc |
| heta hdelta hzeta hs hlow hrange hnorm' hB hleft hcenter hparameter |
| have hmstar : 0 < mstar := by dsimp [mstar]; positivity |
| have ht0 : 0 < theta0 := lt_min (half_pos hmstar) (by positivity) |
| refine ⟨i,d i,hdi,hd i,hmi,hprod,hdub,hli,hri,hBi,ht0,?_⟩ |
| intro theta htheta |
| have hlowtheta : -m i/2 ≤ theta := by |
| have HH : theta0 ≤ mstar/2 := min_le_left _ _ |
| linarith only [htheta.1,HH,hmi] |
| have hthetaa : -a0/3 ≤ theta := by |
| have HH : theta0 ≤ a0/3 := min_le_right _ _ |
| linarith only [htheta.1,HH] |
| have Hconv := barycenter_opposite_extension pj m (fun j => (hm j).le) hsum i hlowtheta htheta.2 |
| rw [hbar] at Hconv |
| have hpie : pj i-p=d i • e := by rw [hd i]; module |
| have hee : inner ℝ e e=1 := by rw [real_inner_self_eq_norm_sq,he]; norm_num |
| have Hrad := outward_radial_ratio hq ha0 hdi hqa hdub hthetaa |
| refine ⟨Hconv,?_,?_⟩ |
| · simpa only [hpie,inner_add_left,real_inner_smul_left,hee,mul_one] using Hrad.1 |
| · have Hn : ‖pj i-p‖^2=d i^2 := by |
| rw [hpie,norm_smul,he,mul_one,Real.norm_eq_abs,sq_abs] |
| have Hi : inner ℝ (p+theta • (pj i-p)) (pj i-p)= |
| (inner ℝ p e+theta*d i)*d i := by |
| rw [hpie] |
| simp only [inner_add_left,inner_smul_right,real_inner_smul_left,hee] |
| ring |
| rw [Hn,Hi] |
| exact Hrad.2 |
|
|
| end OutwardGeometry |
| end WeakMTWTransport |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
