| import OAI.Geometry.Convex.GeneralMahler.MixedField |
|
|
| namespace OAI |
| /-! |
| |
| -/ |
| noncomputable section |
| open Set Filter Real Matrix MeasureTheory MeasureTheory.Measure |
| open scoped Topology ENNReal NNReal RealInnerProductSpace Matrix.Norms.L2Operator MatrixOrder |
| namespace GeneralMahler |
| open Layers |
| variable {m : ℕ} |
|
|
| namespace ProjField |
| variable (q : ProjField m) |
|
|
| def kerH (f : ℝ → ℝ) (z : ℝ) (x : Rn m) := deriv f z • q.PD z x |
| def Hmat (f : ℝ → ℝ) (x : Rn m) := ∫ z, q.kerH f z x |
| def Amat := q.Hmat id |
| def M (i : Fin m) := ∫ x, x i • q.Amat x ∂normal m |
| def Lmat (x : Rn m) := ∑ i, x i • q.M i |
| def Rmat (x : Rn m) := q.Amat x - q.Lmat x |
| def avgA := ∫ x, q.Amat x ∂normal m |
| lemma PD_herm (z x) : (q.PD z x).IsHermitian := |
| ((Matrix.isHermitian_one).smul (show IsSelfAdjoint (p z) by simp [IsSelfAdjoint])).sub (q.Pmat_herm _ _) |
|
|
| variable [NeZero m] {f : ℝ→ℝ} (h : PolyBound (deriv f)) |
|
|
| lemma H_mixed (h : PolyBound (deriv f)) : mixed (q.kerH f) := |
| q.mixed_PD.product |
| (g := fun z (_ : Rn m) => deriv f z) |
| (h.comp PolyBound.fst) (fun x y => by |
| rw [kerH,norm_smul]; exact le_of_eq (mul_comm ..)) |
| |
| omit [NeZero m] in |
| lemma H_sm : StronglyMeasurable (q.kerH f).uncurry := |
| ((measurable_deriv f).stronglyMeasurable.comp_measurable measurable_fst).smul q.PD_sm |
|
|
| omit [NeZero m] in |
| lemma H_ams (y : Rn m) : |
| AEStronglyMeasurable (fun x => q.kerH f x y) := |
| ((q.H_sm (f := f)).comp_measurable (measurable_id.prodMk measurable_const)).aestronglyMeasurable |
|
|
| include h in |
| lemma H_poly : PolyBound (q.Hmat f) := |
| poly_integral_left (q.H_mixed h) (q.H_ams) |
|
|
| include h in |
| lemma H_left_int (y) : Integrable (fun x => q.kerH f x y) := |
| mixed_integrable_left (q.H_mixed h) y (q.H_ams y) |
|
|
| omit [NeZero m] in |
| lemma H_SM : StronglyMeasurable (q.Hmat f) := (q.H_sm).integral_prod_left |
|
|
| include h in |
| lemma H_integrable : Integrable (q.Hmat f) (normal m) := |
| (q.H_poly h).gaussian_integrable q.H_SM.aestronglyMeasurable |
|
|
| include h in |
| lemma H_hermitian (x) : (q.Hmat f x).IsHermitian := |
| hermitian_integral (q.H_left_int h _) |
| (ae_of_all _ fun z => (q.PD_herm ..).smul (show IsSelfAdjoint (deriv f z) by simp [IsSelfAdjoint])) |
|
|
| lemma poly_id_der : PolyBound (deriv (id : ℝ → ℝ)) := by |
| rw [show deriv (id : ℝ → ℝ) = (fun _ => (1:ℝ)) from funext fun x => deriv_id x] ; exact PolyBound.const 1 |
| |
| lemma A_poly : PolyBound q.Amat := q.H_poly poly_id_der |
| omit [NeZero m] in |
| lemma A_SM : StronglyMeasurable q.Amat := q.H_SM |
| lemma A_integrable : Integrable q.Amat (normal m) := q.H_integrable poly_id_der |
|
|
| def coord (i : Fin m) : Rn m →L[ℝ] ℝ := (ContinuousLinearMap.proj i).comp |
| (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin m => ℝ)).toContinuousLinearMap |
| omit [NeZero m] in |
| @[simp] lemma coord_apply (i : Fin m) (x) : coord i x = x i := rfl |
|
|
| lemma A_coord_poly (i : Fin m) : PolyBound (fun x : Rn m => (x i) • q.Amat x) := |
| (PolyBound.clm (coord i)).smul q.A_poly |
|
|
| lemma A_coord_int (i : Fin m) : |
| Integrable (fun x : Rn m => x i • q.Amat x) (normal m) := by |
| apply (q.A_coord_poly i).gaussian_integrable |
| exact (((coord i).continuous.stronglyMeasurable).smul q.A_SM).aestronglyMeasurable |
|
|
| lemma avgA_sym : q.avgA.IsHermitian := hermitian_integral q.A_integrable |
| (ae_of_all _ fun _z => q.H_hermitian poly_id_der _) |
|
|
| lemma M_sym (i) : (q.M i).IsHermitian := |
| hermitian_integral (q.A_coord_int i) (ae_of_all _ fun z => (q.H_hermitian poly_id_der _).smul (by simp [IsSelfAdjoint])) |
|
|
| omit [NeZero m] in |
| lemma L_linear : ∃ l : Rn m →L[ℝ] Mat m, (l : Rn m → Mat m) = q.Lmat := by |
| classical |
| use ∑ i, (coord i).smulRight (q.M i) |
| ext x; simp [Lmat] |
| |
| omit [NeZero m] in |
| lemma L_poly : PolyBound q.Lmat := by |
| obtain ⟨l,hl⟩ := q.L_linear; rw [← hl]; exact PolyBound.clm _ |
| |
| omit [NeZero m] in |
| lemma L_cont : Continuous q.Lmat := by |
| obtain ⟨l,hl⟩ := q.L_linear; rw [← hl]; exact l.continuous |
| |
| lemma L_sym (x) : (q.Lmat x).IsHermitian := |
| by |
| unfold Matrix.IsHermitian Lmat |
| rw [conjTranspose_sum]; exact Finset.sum_congr rfl fun i _ => (q.M_sym i).smul (by simp [IsSelfAdjoint]) |
| |
| end ProjField |
| end GeneralMahler |
| |
| end |
| |
| end OAI |
| |
