| import OAI.Geometry.Convex.GeneralMahler.ProfileGrowth |
| import OAI.Geometry.Convex.GeneralMahler.ScatterField |
| import OAI.Geometry.Convex.GeneralMahler.LayerMeasure |
|
|
| namespace OAI |
| /-! μ functionals and integrable density algebra for Section §03-04. -/ |
| noncomputable section |
| open Set Filter MeasureTheory MeasureTheory.Measure Matrix Metric Real |
| open scoped ENNReal NNReal Topology MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace |
| namespace GeneralMahler |
| open Layers Profile |
|
|
| namespace rapid |
| variable {f g:ℝ→ℝ} |
| lemma rfadd (hf:rapid f) (hg:rapid g) : rapid fun x=> f x+g x := by |
| intro n |
| obtain ⟨C,hc,h⟩ := hf n |
| obtain ⟨D,hd,h'⟩ := hg n |
| refine ⟨C+D, by positivity, fun x=>?_⟩ |
| apply le_trans (mul_le_mul_of_nonneg_right (norm_add_le ..) (by positivity)) |
| rw [add_mul]; exact add_le_add (h x) (h' x) |
| lemma rfneg (hf:rapid f) : rapid fun x=> -f x := hf.mono (by simp) |
| lemma rfsub (hf:rapid f) (hg:rapid g) : rapid fun x=> f x-g x := by |
| simpa only [sub_eq_add_neg] using hf.rfadd hg.rfneg |
| end rapid |
| lemma poly_st : PolyBound st := PolyBound.of_bound 1 fun x=> by unfold st; split <;> simp |
| lemma poly_p : PolyBound p := PolyBound.of_bound 1 fun x=>by |
| rw [Real.norm_of_nonneg (p_pos x).le]; exact (p_lt_one x).le |
| lemma rapid_half_a : rapid fun x=> a x*(1-st x) := by |
| apply rapid_phi.mono |
| intro x |
| unfold st; split_ifs with h |
| · simp |
| rw [sub_zero,mul_one,Real.norm_of_nonneg (a_pos _).le, |
| Real.norm_of_nonneg (phi_pos _).le] |
| simpa only [neg_neg,neg_phi] using a_high (-x) (by linarith) |
| lemma rapid_ar : rapid fun x=> a (-x)*st x := by |
| apply rapid_phi.mono |
| intro x |
| unfold st; split_ifs with h |
| · rw [mul_one,Real.norm_of_nonneg (a_pos _).le, |
| Real.norm_of_nonneg (phi_pos _).le]; exact a_high x h |
| simp |
|
|
| -- integrable densities with polynomial weights |
| def MomentF (f:ℝ→ℝ) := |
| ∀ g : ℝ→ℝ, Measurable g→PolyBound g→Integrable fun x=>f x*g x |
| namespace MomentF |
| variable {f g:ℝ→ℝ} |
| lemma of_rapid (hf:rapid f) (hm:Measurable f) : MomentF f := |
| fun _g hg h=> (hf.product h).integrable_real (hm.mul hg) |
| lemma inte (hf:MomentF f) : Integrable f := by simpa using hf (fun _ => 1) measurable_const (PolyBound.const _) |
| lemma add (hf:MomentF f) (hg:MomentF g) : MomentF fun x=>f x+g x := by |
| intro u hu h; simp_rw [add_mul] |
| exact (hf u hu h).add (hg u hu h) |
| lemma neg (hf:MomentF f) : MomentF fun x=> -f x := by |
| intro u hu h |
| simp_rw [neg_mul]; exact (hf u hu h).neg |
| lemma sub (hf:MomentF f) (hg:MomentF g) : MomentF fun x=> f x-g x := by simpa only [sub_eq_add_neg] using hf.add hg.neg |
| lemma mul (hf:MomentF f) (hg:PolyBound g) (hm:Measurable g) : MomentF fun x=>f x*g x := by |
| intro u hu h |
| simp_rw [mul_assoc]; exact hf _ (hm.mul hu) (hg.mul h) |
| lemma mono (hf:MomentF f) (hm:Measurable g) (h:∀ x,‖g x‖ ≤ ‖f x‖) : |
| MomentF g := by |
| intro u hu h' |
| apply (hf u hu h').mono (hm.mul hu).aestronglyMeasurable |
| exact ae_of_all _ fun x=>by rw [Pi.mul_apply,norm_mul,norm_mul]; exact mul_le_mul_of_nonneg_right (h x) (norm_nonneg _) |
| end MomentF |
| variable {m:ℕ} [NeZero m] |
| namespace ProjField |
| variable (q:ProjField m) |
|
|
| -- deltas |
| def dB (x:ℝ) := q.Bt x-q.s0*a (-x) |
| def dr (x:ℝ) := q.r1 x-q.s0*(1-p x) |
| def hDel (x:ℝ) := q.avH x-q.s0*p x |
| def dm (x:ℝ) := q.layerW x-q.s0*phi x |
| def delt (j:ℝ→ℝ) := |
| -(∫ x, (2*p x*q.dB x*j x + (p x*q.dr x+q.hDel x)*(2*x*j x-deriv j x))) |
|
|
| lemma Bt_poly : PolyBound q.Bt := by |
| have he : q.Bt = fun z=> ((m:ℝ)⁻¹)*⟪q.U,a z • q.U-q.shift z⟫ := by |
| ext x; rw [Bt,q.Y_mean]; ring |
| rw [he] |
| let l := innerSL ℝ q.U |
| exact (PolyBound.const _).mul ((PolyBound.clm (show Rn m→L[ℝ]ℝ from l)).comp |
| ((poly_a.smul (PolyBound.const _)).sub q.s_poly)) |
|
|
| lemma B_tail : rapid fun z=> q.Bt z* st z := by |
| let f := fun z x=> st z • q.YT z x |
| let v := q.dualMap |
| have hy : PolyBound q.YT.uncurry := by |
| change PolyBound (fun u:ℝ×Rn m=>coneProj q.D (-(q.Z u.2+q.shift u.1))) |
| have hp : PolyBound (fun u:ℝ×Rn m=> -(q.Z u.2+q.shift u.1)) := |
| (((PolyBound.clm q.root.toContinuousLinearMap).comp PolyBound.snd).add |
| (q.s_poly.comp PolyBound.fst)).neg |
| exact (PolyBound.lipschitz (coneProj_lip q.D)).comp hp |
| have hh : mixed f := by |
| apply mixed.of_cutoff (show PolyBound f.uncurry from |
| (poly_st.comp PolyBound.fst).smul hy) |
| (show PolyBound (fun x:Rn m=>v.M*(1+‖x‖)) from |
| (PolyBound.const _).mul ((PolyBound.const _).add PolyBound.id.norm)) |
| intro z x hz |
| unfold f st; split_ifs with h |
| · rw [Real.norm_of_nonneg h] at hz |
| have hi := v.zero_y (-z) x (by simpa using hz) |
| rw [show v.y (-z)= _ from q.Y_sc (-z),neg_neg] at hi |
| simp only [hi,smul_zero] |
| simp |
| let l := ((m:ℝ)⁻¹) • (innerSL ℝ q.U) |
| have hl (x:Rn m) : l x=⟪q.U,x⟫/(m:ℝ) := by unfold l; simp [div_eq_inv_mul] |
| have hi (z:ℝ) : Integrable (f z) (normal m) := (q.Y_int z).smul (st z) |
| have he : (fun z=> q.Bt z*st z) = fun z=> ∫ x,l (f z x) ∂normal m := by |
| ext z |
| rw [l.integral_comp_comm (hi _)] |
| unfold f; rw [integral_smul,_root_.map_smul,hl,Bt] |
| rw [smul_eq_mul,mul_comm] |
| rw [he] |
| apply mixed_integral_right _ fun z=>(l.integrable_comp (hi z)).aestronglyMeasurable |
| let c := fun (_:ℝ) (_:Rn m)=> ‖l‖ |
| apply hh.product (show PolyBound c.uncurry from PolyBound.const _) |
| intro x y |
| apply (l.le_opNorm _).trans |
| change ‖l‖*_ ≤ _*‖‖l‖‖ |
| rw [Real.norm_of_nonneg (norm_nonneg l),mul_comm] |
|
|
| lemma dB_poly : PolyBound q.dB := q.Bt_poly.sub |
| ((PolyBound.const _).mul (poly_a.comp PolyBound.id.neg)) |
|
|
| lemma dB_tail : rapid fun z=> q.dB z*st z := by |
| convert q.B_tail.rfsub (rapid_ar.product (PolyBound.const q.s0)) using 1 |
| ext x; unfold dB; ring |
| lemma pB_tail : rapid fun z=> p z*q.dB z := by |
| convert ((rapid_p.product q.dB_poly).rfadd q.dB_tail) using 1 |
| ext x; ring |
| lemma aB_tail : rapid fun z=> a z*q.dB z := by |
| convert (rapid_half_a.product q.dB_poly).rfadd (q.dB_tail.product poly_a) using 1 |
| ext x; ring |
| lemma hd_tail : rapid q.hDel := by |
| convert q.tail_H.rfsub (rapid_p.product (PolyBound.const q.s0)) using 1 |
| ext x; unfold hDel; ring |
|
|
| lemma moment_w : MomentF q.layerW := by |
| intro f hm h |
| obtain ⟨C,n,hc,h⟩ := h |
| apply ((q.w_moment n).mul_const C).mono' (q.w_cont.measurable.mul hm).aestronglyMeasurable |
| apply ae_of_all |
| intro x |
| rw [Pi.mul_apply,norm_mul,Real.norm_of_nonneg (q.w_positive x).le] |
| apply le_trans (mul_le_mul_of_nonneg_left (h x) (q.w_positive x).le) |
| exact le_of_eq (by ring) |
| lemma ar_moment : MomentF fun x=>a x*q.r1 x := |
| q.moment_w.mono (ca.measurable.mul q.Bt_cd.continuous_deriv_one.measurable.neg) (fun x=>by |
| rw [Real.norm_of_nonneg (mul_nonneg (a_pos x).le (q.r1_pos x).le), |
| Real.norm_of_nonneg (q.w_positive x).le] |
| exact le_add_of_nonneg_left (mul_nonneg (p_pos x).le (q.Bt_nonneg x))) |
| lemma a0_id (x:ℝ) : p x*a (-x)+a x*(1-p x)=phi x := by |
| simp only [a,neg_phi,neg_p]; ring |
| lemma arDelta_moment : MomentF fun x=> a x*q.dr x := by |
| have hf : MomentF (fun x=>a x*(1-p x)) := |
| (MomentF.of_rapid rapid_phi c_phi.measurable).mono |
| (ca.measurable.mul (measurable_const.sub cp.measurable)) (fun x=> by |
| rw [Real.norm_of_nonneg (mul_pos (a_pos _) (q_pos _)).le, |
| Real.norm_of_nonneg (phi_pos _).le] |
| exact (le_add_of_nonneg_left (mul_nonneg (p_pos x).le (a_pos (-x)).le)).trans_eq (a0_id _)) |
| convert q.ar_moment.sub (hf.mul (PolyBound.const q.s0) measurable_const) using 1 |
| ext x; unfold dr; ring |
| lemma prDelta_moment : MomentF fun x=>p x*q.dr x := by |
| convert q.arDelta_moment.mul ratio_bound (cp.measurable.div ca.measurable) using 1 |
| ext x; field_simp [ne_of_gt (a_pos x)] |
| end ProjField |
| end GeneralMahler |
|
|
| end |
|
|
| end OAI |
|
|
