| import OAI.Geometry.Convex.GeneralMahler.TraceOrder |
| import OAI.Geometry.Convex.GeneralMahler.MixedField |
| import OAI.Geometry.Convex.GeneralMahler.MillsLog |
|
|
| namespace OAI |
| /-! Deterministic-endcut layer of PSD contractions. Carries the |
| precise data needed for log-Jacobian estimate. -/ |
| noncomputable section |
| open MeasureTheory MeasureTheory.Measure Filter Set Matrix Real Metric |
| open scoped Topology NNReal ENNReal MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace |
| namespace GeneralMahler |
| open Layers |
| variable {m:ℕ} |
| structure Edge (m:ℕ) where |
| P : ℝ → Rn m → Mat m |
| psd : ∀ z x, 0 ≤ P z x |
| le_one : ∀ z x,P z x≤1 |
| meas : Measurable P.uncurry |
| C : ℝ |
| hp : 1 ≤ C |
| tend_pos : ∀ z x,C*(1+‖x‖)<z → P z x=1 |
| tend_neg : ∀ z x,C*(1+‖x‖)< -z → P z x=0 |
|
|
| variable [NeZero m] |
| namespace ProjField |
| variable (q:ProjField m) |
| lemma Field_tend : |
| ∃ c≥(1:ℝ), (∀ z x,c*(1+‖x‖)<z → q.Pmat z x=1) ∧ |
| (∀ z x,c*(1+‖x‖)< -z → q.Pmat z x=0) := by |
| obtain ⟨K,hk⟩ := q.Field_isBound |
| obtain ⟨a,ha,h⟩ := positive_cutoff m hk.one_le |
| obtain ⟨b,hb,h'⟩ := negative_cutoff m hk.one_le |
| refine ⟨max a b,ha.trans (le_max_left ..),?_,?_⟩ |
| · intro z x hx; apply projJac_id q.C (h _ hk _ x ?_) |
| exact lt_of_le_of_lt (mul_le_mul_of_nonneg_right (le_max_left ..) (by positivity)) hx |
| intro z x hx |
| have hi := h' q hk (-z) x (lt_of_le_of_lt |
| (mul_le_mul_of_nonneg_right (le_max_right ..) (by positivity)) hx) |
| rw [neg_neg] at hi |
| exact projJac_zero q.C hi |
|
|
| def edge : Edge m where |
| P := q.Pmat |
| psd := q.Pmat_psd |
| le_one := q.Pmat_le |
| meas := |
| ((strongly_projJac q.C).comp_measurable |
| ((q.root.continuous.comp continuous_snd).add (q.shift_cd.continuous.comp continuous_fst)).measurable).measurable |
| C := q.Field_tend.choose |
| hp := q.Field_tend.choose_spec.1 |
| tend_pos := q.Field_tend.choose_spec.2.1 |
| tend_neg := q.Field_tend.choose_spec.2.2 |
| end ProjField |
| namespace Edge |
| variable (e:Edge m) |
| def ref : Edge m where |
| P := fun z x=>1-e.P (-z) x |
| psd := fun z x=> sub_nonneg.mpr (e.le_one ..) |
| le_one := fun z x=> sub_le_self _ (e.psd ..) |
| meas := measurable_const.sub (e.meas.comp ((measurable_neg.comp measurable_fst).prodMk measurable_snd)) |
| C := e.C |
| hp := e.hp |
| tend_pos := fun z x hx=> by rw [e.tend_neg _ _ (by simpa using hx),sub_zero] |
| tend_neg := fun z x hx=> by rw [e.tend_pos _ _ hx,sub_self] |
|
|
| def radius (x:Rn m) := e.C*(1+‖x‖) |
| omit [NeZero m] in |
| lemma r_meas : Continuous e.radius := by unfold radius; fun_prop |
| omit [NeZero m] in |
| lemma r_poly : PolyBound e.radius := (PolyBound.const _).mul |
| ((PolyBound.const _).add PolyBound.id.norm) |
| omit [NeZero m] in |
| lemma r_pos (x) : 0 < e.radius x := mul_pos (lt_of_lt_of_le zero_lt_one e.hp) (by positivity) |
| omit [NeZero m] in |
| lemma Pcont (x y) : ‖e.P x y‖ ≤ 1 := by |
| apply spectrum_norm_box zero_le_one (e.psd ..).posSemidef.1 |
| · apply le_trans _ (e.psd ..) |
| have he := scalar_le' (m:=m) (-1) 0 (by norm_num) |
| simpa only [scalar, zero_smul] using he |
| simpa [scalar] using e.le_one x y |
| omit [NeZero m] in |
| lemma P_poly : PolyBound e.P.uncurry := PolyBound.of_bound 1 fun ⟨x,y⟩=>e.Pcont x y |
|
|
| def EP (z:ℝ) (x:Rn m) := e.P z x-st z • (1:Mat m) |
| lemma st_poly : PolyBound st := PolyBound.of_bound 1 fun x=>by unfold st; split <;> simp |
| omit [NeZero m] in |
| lemma EP_mixed : mixed e.EP := by |
| have h : PolyBound (fun x:ℝ × Rn m=> st x.1 • (1:Mat m)) := (st_poly.comp PolyBound.fst).smul |
| (PolyBound.const _) |
| apply mixed.of_cutoff (e.P_poly.sub h) e.r_poly |
| intro x y hx |
| unfold EP st; split_ifs with ht |
| · rw [e.tend_pos _ _ (by rw [Real.norm_of_nonneg ht] at hx; exact hx),one_smul,sub_self] |
| rw [e.tend_neg _ _ (by rw [Real.norm_eq_abs,abs_of_neg (lt_of_not_ge ht)] at hx;exact hx),zero_smul,sub_self] |
| |
| def WP (z:ℝ) (x:Rn m) := MillsW z • e.P z x |
| omit [NeZero m] in |
| lemma wp_meas : StronglyMeasurable e.WP.uncurry := |
| ((c_mw.measurable.comp measurable_fst).smul e.meas).stronglyMeasurable |
| omit [NeZero m] in |
| lemma wp_psd (z x) : 0 ≤ e.WP z x := |
| smul_nonneg (mw_pos z).le (e.psd ..) |
| omit [NeZero m] in |
| lemma wp_le (z x) : e.WP z x≤MillsW z•1 := |
| smul_le_smul_of_nonneg_left (e.le_one ..) (mw_pos z).le |
| lemma WP_mixed : mixed e.WP := by |
| let f := fun z (x:Rn m)=> (MillsW z*st z) • (1:Mat m) |
| let g := fun z x => MillsW z • e.EP z x |
| have hh : e.WP = fun z x=> f z x+ g z x := by |
| funext z x; simp only [WP,f,g,EP,mul_smul,smul_sub]; abel |
| rw [hh] |
| have hi : mixed f := (mixed.of_rapid (Y:=Rn m) rapid_mws).mono |
| (fun z x=>by unfold f; rw [norm_smul]; simp) |
| apply hi.add |
| let H := fun z (_:Rn m) => MillsW z |
| have h : PolyBound H.uncurry := poly_mw.comp PolyBound.fst |
| exact e.EP_mixed.product h (fun z x=>by unfold g H; rw [norm_smul,mul_comm]) |
| omit [NeZero m] in |
| lemma wp_slice_SM (x) : StronglyMeasurable (fun z=> e.WP z x) := |
| e.wp_meas.comp_measurable (measurable_id.prodMk measurable_const) |
| lemma wp_slice_i (x) : Integrable (fun z=>e.WP z x) := |
| mixed_integrable_left e.WP_mixed x (e.wp_slice_SM x).aestronglyMeasurable |
| lemma wp_int : Integrable e.WP.uncurry (volume.prod (normal m)) := |
| mixed_integrable e.WP_mixed e.wp_meas.aestronglyMeasurable |
|
|
| def Jedge (x:Rn m) := ∫ z,e.WP z x |
| omit [NeZero m] in |
| lemma Jedge_SM : StronglyMeasurable e.Jedge := e.wp_meas.integral_prod_left |
| lemma Jedge_poly : PolyBound e.Jedge := poly_integral_left e.WP_mixed fun x=> (e.wp_slice_SM x).aestronglyMeasurable |
| lemma Jedge_bound (x) : scalar m (MillsC (e.radius x)) ≤ e.Jedge x := by |
| let a := e.radius x |
| have hf := integral_add_compl measurableSet_Iic (e.wp_slice_i x) (s:=Iic a) |
| rw [compl_Iic] at hf |
| have hh : (∫ z in Ioi a,e.WP z x) = scalar m (MillsC a) := by |
| rw [← (mw_tail a).2,scalar,← integral_smul_const] |
| exact setIntegral_congr_fun measurableSet_Ioi fun z hz=>by rw [WP,e.tend_pos _ _ hz] |
| have hi : (0:Mat m) ≤ ∫ z in Iic a,e.WP z x := |
| (psd_integral (e.wp_slice_i x).integrableOn (ae_of_all _ fun z=>(e.wp_psd z x).posSemidef)).nonneg |
| unfold Jedge |
| rw [← hf,hh]; exact le_add_of_nonneg_left hi |
| lemma Jedge_pd (x) : (e.Jedge x).PosDef := by |
| let B := scalar m (MillsC (e.radius x)) |
| have hi : B.PosDef := form_posDef (mc_pos _) |
| (show B∈specBox m (MillsC (e.radius x)) (MillsC (e.radius x)) from ⟨scalar_sym ..,le_rfl,le_rfl⟩) |
| have hh := Matrix.le_iff.mp (e.Jedge_bound x) |
| have h := hi.add_posSemidef hh |
| simpa only [B, add_sub_cancel] using h |
|
|
| end Edge |
| end GeneralMahler |
|
|
| end |
|
|
| end OAI |
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