| import OAI.Geometry.Convex.GeneralMahler.Hermite.Pair |
| import OAI.Geometry.Convex.GeneralMahler.Entropy |
|
|
| namespace OAI |
| /-! The centered fields W_f from §02,04. We use coefficient |
| derivatives (Hermite) as justified by differentiation by parts: |
| rather than interchanging weak derivatives, integrate the coefficient |
| identity using the layer cutoff bounds. -/ |
| noncomputable section |
| open MeasureTheory MeasureTheory.Measure Filter Set Matrix Real Metric |
| open scoped Topology NNReal ENNReal MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace |
| namespace GeneralMahler |
| open Profile Layers HMode |
| variable {m:ℕ} [NeZero m] |
| namespace ProjField |
| variable (q:ProjField m) |
| def Xb (z:ℝ) (x:Rn m) := (1-st z) • q.XT z x |
| def Yb (z:ℝ) (x:Rn m) := st z • q.YT z x |
| lemma XPbound : PolyBound q.XT.uncurry := by |
| have h := q.primalMap.yp |
| simp_rw [show q.primalMap.y=q.XT from funext q.X_sc] at h; exact h |
| lemma YPbound : 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 |
|
|
| lemma XbM : mixed q.Xb := by |
| let v := q.primalMap |
| apply mixed.of_cutoff (show PolyBound q.Xb.uncurry from |
| (((PolyBound.const (1:ℝ)).sub poly_st).comp PolyBound.fst).smul q.XPbound) |
| (show PolyBound (fun x:Rn m=> v.M*(1+‖x‖)) from |
| (PolyBound.const _).mul ((PolyBound.const _).add PolyBound.id.norm)) |
| intro z x hx |
| unfold Xb st; split_ifs with h |
| · simp |
| rw [Real.norm_eq_abs,abs_of_neg (lt_of_not_ge h)] at hx |
| have hi := v.zero_y z x hx |
| rw [show v.y z= _ from q.X_sc _] at hi |
| simp only [hi,smul_zero] |
| lemma YbM : mixed q.Yb := by |
| let v := q.dualMap |
| apply mixed.of_cutoff (show PolyBound q.Yb.uncurry from |
| (poly_st.comp PolyBound.fst).smul q.YPbound) |
| (show PolyBound (fun x:Rn m=> v.M*(1+‖x‖)) from |
| (PolyBound.const _).mul ((PolyBound.const _).add PolyBound.id.norm)) |
| intro z x hx |
| unfold Yb st; split_ifs with h |
| · rw [Real.norm_of_nonneg h] at hx |
| have hi := v.zero_y (-z) x (by simpa using hx) |
| rw [show v.y (-z)= _ from q.Y_sc _,neg_neg] at hi |
| simp only [hi,smul_zero] |
| simp |
|
|
| def WW (z:ℝ) (x:Rn m) := |
| p z • q.Z x-q.XT z x + a z • q.U |
|
|
| omit [NeZero m] in |
| lemma WW_mean (z:ℝ) : (∫ x,q.WW z x ∂normal m)=0 := by |
| have he : Integrable q.Z (normal m) := by |
| have hZ : PolyBound q.Z := PolyBound.clm q.root.toContinuousLinearMap |
| exact hZ.gaussian_integrable q.root.continuous.aestronglyMeasurable |
| unfold WW |
| have hi : Integrable (fun x=> p z • q.Z x) (normal m) := he.smul (p z) |
| rw [integral_add,integral_sub hi (q.X_int z), q.XT_mean, integral_smul] |
| · have he : (∫ x,q.Z x ∂normal m)=0 := by |
| have h := affineN_mean q.root 0 |
| simpa only [affineN,Z,add_zero] using h |
| rw [he]; simp |
| · exact hi.sub (q.X_int z) |
| exact integrable_const _ |
| |
| omit [NeZero m] in |
| lemma WW_eq (z x) : |
| q.WW z x=(p z-st z) • q.Z x-q.Xb z x -q.Yb z x+ |
| (∫ y,q.Xb z y ∂normal m)+(∫ y,q.Yb z y ∂normal m) := by |
| unfold WW Xb Yb |
| rw [integral_smul,integral_smul,q.XT_mean,q.Y_mean,q.YT_sub] |
| module |
| lemma WW_m : mixed q.WW := by |
| let f := fun z x=> (p z-st z) • q.Z x |
| have hp : PolyBound q.Z := PolyBound.clm q.root.toContinuousLinearMap |
| have hf : mixed f := (mixed.of_rapid (Y:=Rn m) rapid_p).product (g:=fun _ x=>q.Z x) |
| (show PolyBound (fun u:ℝ×Rn m=> q.Z u.2) from hp.comp PolyBound.snd) (fun x y=>by rw [show f x y= _ from rfl,norm_smul]) |
| have hx := mixed_integral_right q.XbM fun z=> (show Integrable (q.Xb z) (normal m) |
| from (q.X_int z).smul (1-st z)).aestronglyMeasurable |
| have hy := mixed_integral_right q.YbM fun z=> (show Integrable (q.Yb z) (normal m) |
| from (q.Y_int z).smul (st z)).aestronglyMeasurable |
| rw [show q.WW= _ from funext fun z=>funext fun x=>q.WW_eq z x] |
| exact (((hf.sub q.XbM).sub q.YbM).add (mixed.of_rapid hx)).add (mixed.of_rapid hy) |
|
|
| omit [NeZero m] in |
| lemma WW_c : Continuous q.WW.uncurry := |
| (((cp.comp continuous_fst).smul (q.root.continuous.comp continuous_snd)).sub q.meas_XT).add |
| ((ca.comp continuous_fst).smul continuous_const) |
| def WK (f:ℝ→ℝ) (z:ℝ) (x:Rn m) := deriv f z • q.WW z x |
| def WH (f:ℝ→ℝ) (x:Rn m) := ∫ z,q.WK f z x |
|
|
| omit [NeZero m] in |
| lemma WK_SM (f) : StronglyMeasurable (q.WK f).uncurry := |
| (((measurable_deriv f).comp measurable_fst).stronglyMeasurable).smul q.WW_c.stronglyMeasurable |
| lemma WK_m {f} (hf:TestF f) : mixed (q.WK f) := by |
| let g := fun (z:ℝ) (_:Rn m)=> deriv f z |
| apply q.WW_m.product (show PolyBound g.uncurry from hf.der.poly.comp PolyBound.fst) |
| intro z x |
| unfold WK g; rw [norm_smul,mul_comm] |
| lemma WH_reg {f} (hf:TestF f) : regular (q.WH f) := |
| ⟨poly_integral_left (q.WK_m hf) (fun _x=> (((q.WK_SM f).comp_measurable |
| (measurable_id.prodMk measurable_const)).aestronglyMeasurable)), |
| (q.WK_SM f).integral_prod_left.aestronglyMeasurable⟩ |
| lemma WH_mean {f} (hf:TestF f) : (∫ x,q.WH f x ∂normal m)=0 := by |
| unfold WH |
| rw [integral_integral_swap] |
| · unfold WK; simp_rw [integral_smul,q.WW_mean]; simp |
| exact (mixed_integrable (μ:=volume) (ν:=normal m) (q.WK_m hf) |
| (q.WK_SM f).aestronglyMeasurable).swap |
|
|
| def act (i:Fin m) : Mat m→L[ℝ]Rn m := |
| (ContinuousLinearMap.apply ℝ (Rn m) (op q.S (HMode.e i))).comp opCLE.toContinuousLinearMap |
| omit [NeZero m] in |
| lemma act_eq (i) (A:Mat m) : q.act i A=op A (op q.S (HMode.e i)) := rfl |
|
|
| omit [NeZero m] in |
| lemma WWd (z:ℝ) : |
| ∀ᵐ x∂normal m, HasFDerivAt (q.WW z) |
| ((op (q.PD z x)).comp (op q.S)) x := by |
| filter_upwards [q.XT_diff z] with x hx |
| have hf := ((((q.root.hasFDerivAt).const_smul (p z)).sub hx).add_const (Layers.a z • q.U)) |
| have hl : (op (q.PD z x)).comp (op q.S)=p z•q.root.toContinuousLinearMap - (op (q.Pmat z x)).comp (op q.S) := by |
| unfold PD |
| rw [q.matrix_eq] |
| change op (p z • 1 - _)*op q.S=p z•op q.S-op (q.Pmat z x)*op q.S |
| rw [← _root_.map_mul,sub_mul,smul_mul_assoc,one_mul,_root_.map_sub,_root_.map_smul,_root_.map_mul] |
| rw [hl]; exact hf |
| omit [NeZero m] in |
| lemma WW_lip (z:ℝ) : ∃ K,LipschitzWith K (q.WW z) := by |
| obtain ⟨K,hk⟩ := q.XT_lipschitz z |
| exact ⟨_, (((p z • q.root.toContinuousLinearMap).lipschitzWith).sub hk).add (LipschitzWith.const _)⟩ |
| lemma WDer_p (z:ℝ) (i:Fin m) : |
| PolyBound (fun x=> q.act i (q.PD z x)) := by |
| have hh : PolyBound (q.PD z) := |
| (q.mixed_PD.poly).comp (f:=(fun x=>(z,x))) ((PolyBound.const z).prodMk PolyBound.id) |
| exact (PolyBound.clm (q.act i)).comp hh |
| lemma WW_recur (z:ℝ) (i:Fin m) (a:MI m) : |
| cof (fun x=> q.act i (q.PD z x)) a=sn (a i+1) • cof (q.WW z) (inc a i) := by |
| obtain ⟨K,hk⟩ := q.WW_lip z |
| let D := fun x=> (op (q.PD z x)).comp (op q.S) |
| have hm : AEStronglyMeasurable (fun x=>q.act i (q.PD z x)) (normal m) := |
| ((q.act i).continuous.comp_stronglyMeasurable (q.PD_sm.comp_measurable |
| (measurable_const.prodMk measurable_id))).aestronglyMeasurable |
| have ht : (fun x=>D x (HMode.e i))=(fun x=>q.act i (q.PD z x)) := by funext x; rfl |
| have hi := coef_raise (PolyBound.lipschitz hk) (PLip.lip hk) hk.continuous (D:=D) (q.WWd z) i |
| (ht.symm ▸ q.WDer_p z i) (ht.symm ▸ hm) a |
| rw [ht] at hi; exact hi |
| |
| -- Linear coefficient interchange with layer integral, natural powers |
| omit [NeZero m] in |
| lemma cof_layer {F:Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] |
| {f:ℝ→Rn m→F} (hf:mixed f) (hm:StronglyMeasurable f.uncurry) (a:MI m) : |
| cof (fun x=>∫ z,f z x) a=∫ z,cof (f z) a := by |
| let g := fun z x=> MM a x • f z x |
| have hh : StronglyMeasurable g.uncurry := |
| ((M_cont a).stronglyMeasurable.comp_measurable measurable_snd).smul hm |
| have hg : mixed g := hf.product (g:=fun _ x=>MM a x) |
| (show PolyBound (fun u:ℝ × Rn m=>MM a u.2) from (M_poly a).comp PolyBound.snd) |
| (fun z x=>by unfold g; rw [norm_smul,mul_comm]) |
| unfold cof; simp_rw [← integral_smul] |
| exact integral_integral_swap (mixed_integrable (μ:=volume) (ν:=normal m) hg hh.aestronglyMeasurable).swap |
|
|
| lemma WH_rec {f} (hf:TestF f) (i:Fin m) (a:MI m) : |
| q.act i (cof (q.Hmat f) a)=sn (a i+1) • cof (q.WH f) (inc a i) := by |
| let g := fun z x=> q.act i (q.kerH f z x) |
| have hh (x) : q.act i (q.Hmat f x)=∫ z,g z x := ( (q.act i).integral_comp_comm |
| (q.H_left_int hf.der.poly _)).symm |
| have he : mixed g := (q.H_mixed hf.der.poly).product (g:=fun _ _=>‖q.act i‖) |
| (show PolyBound (fun _:ℝ × Rn m=>‖q.act i‖) from PolyBound.const _) |
| (fun z x=>by unfold g; rw [Real.norm_of_nonneg (norm_nonneg (q.act i)),mul_comm]; apply ContinuousLinearMap.le_opNorm) |
| have hi (z) : cof (g z) a=sn (a i+1) • cof (q.WK f z) (inc a i) := by |
| have h := q.WW_recur z i a |
| have hf (v:Rn m→Rn m) (a:MI m) : cof (fun x=> deriv f z • v x) a=deriv f z • cof v a := by |
| unfold cof; rw [← integral_smul]; congr 1; ext; rw [smul_comm] |
| unfold WK g kerH; simp_rw [_root_.map_smul] |
| rw [hf,hf,h,smul_comm] |
| have hm : regular (q.Hmat f) := ⟨q.H_poly hf.der.poly,q.H_SM.aestronglyMeasurable⟩ |
| rw [cof_clm hm] |
| simp_rw [hh] |
| rw [cof_layer he (((q.act i).continuous).comp_stronglyMeasurable q.H_sm)] |
| simp_rw [hi, integral_smul] |
| rw [show cof (q.WH f) _=_ from cof_layer (q.WK_m hf) (q.WK_SM _) (inc a i)] |
| end ProjField |
| end GeneralMahler |
|
|
| end |
|
|
| end OAI |
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