| import OAI.Geometry.Convex.GeneralMahler.Prim |
| import OAI.Geometry.Convex.GeneralMahler.Compare |
| import OAI.Geometry.Convex.GeneralMahler.MatrixDeriv |
|
|
| namespace OAI |
| /-! Forms on the Hermite coefficients. D and sigma in paper §04. -/ |
| noncomputable section |
| open Set Filter MeasureTheory MeasureTheory.Measure Matrix Real Metric |
| open scoped Topology NNReal ENNReal MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace |
| namespace GeneralMahler |
| open HMode Profile Layers |
| variable {m:ℕ} |
| lemma pj_ss (W A B C D:Mat m) : |
| Pj W (A-B) (C-D)= Pj W A C-Pj W A D-Pj W B C+Pj W B D := by |
| simp_rw [← pJ_eq] |
| simp only [_root_.map_sub,_root_.sub_apply]; ring |
| lemma pJ_smul_left (W:Mat m) (A C:Mat m) (a:ℝ) : |
| Pj W (a•A) C=a*Pj W A C := by simpa using pj_scale a 1 W A C |
| lemma pJ_smul_right (W:Mat m) (A C:Mat m) (a:ℝ) : |
| Pj W A (a•C)=a*Pj W A C := by simpa using pj_scale 1 a W A C |
|
|
| lemma pj_poly_part {f: Rn m→Mat m} (hf:regular f) (W:Mat m) : |
| Integrable (fun x=>Pj W (f x) (f x)) (normal m) := by |
| simp_rw [← pJ_eq] |
| exact ProjField.proj_int W hf hf |
|
|
| namespace ProjField |
| variable [NeZero m] (q:ProjField m) |
| def DLMat (l:ℝ→ℝ) (x:Rn m) (i:Fin m) := fderiv ℝ (q.FL.eval (pl l)) x (e i) |
| def dirF (W:Mat m) (l:ℝ→ℝ) := ∑ i:Fin m, |
| Pt W (fun x=>q.DLMat l x i) (fun x=>q.DLMat l x i) |
| lemma dl_raise {l:ℝ→ℝ} (hl:TestF l) (i:Fin m) (a:MI m) : |
| cof (fun x=>q.DLMat l x i) a=sn (a i+1) • cof (q.FL.eval (pl l)) (inc a i) := by |
| have hh := pl_test hl |
| let f := q.FL.eval (pl l) |
| have hi := q.FL.eval_reg hh |
| have hf : Differentiable ℝ f := fun x=>(q.hasdl x hh).differentiableAt |
| have hv := PLip.fd hf (q.dPoly hh) |
| have h := q.LD_reg i hh |
| exact coef_raise hi.p hv hf.continuous (ae_of_all _ fun x=>(hf x).hasFDerivAt) i h.p h.meas a |
|
|
| lemma Pt_dir (W:Mat m) {l:ℝ→ℝ} (hl:TestF l) : |
| let b := q.FL.eval (pl l) |
| HasSum (fun a:MI m=> (deg a:ℝ)* CpPair W b b a) (q.dirF W l) := by |
| intro b |
| let f := fun (i:Fin m) (a:MI m)=> (a i:ℝ)*CpPair W b b a |
| have he (a): (deg a:ℝ)*CpPair W b b a=∑ i,f i a := by |
| unfold f deg; rw [← Finset.sum_mul,Nat.cast_sum] |
| have hh (i) : |
| let c := fun x=>q.DLMat l x i |
| HasSum (f i) (Pt W c c) := by |
| intro c |
| have hi : regular c := q.LD_reg i (pl_test hl) |
| have hp := pt_sum (B:=W) hi hi |
| have h (a) : CpPair W c c a=f i (inc a i) := by |
| unfold CpPair |
| rw [show cof c a= _ from q.dl_raise hl i a,pj_scale] |
| unfold f CpPair b |
| rw [inc_at,← mul_assoc,← pow_two,snsq] |
| rw [funext h] at hp |
| obtain ⟨he₁,he₂⟩ := inc_sum i (f i) fun a ha=>by simp [ha,f] |
| rw [hp.tsum_eq] at he₂ |
| exact he₂ ▸ (he₁.mpr hp.summable).hasSum |
| simp_rw [he] |
| exact hasSum_sum fun i _=>hh i |
|
|
| def DD (l:ℝ→ℝ) (a:MI m) := kMode a • cof (q.Hmat (fl l)) a-cof (q.FL.eval (pl l)) a |
| def Dterm (l:ℝ→ℝ) (a:MI m) := (1+(deg a:ℝ))*Pj q.covMat (q.DD l a) (q.DD l a) |
| def Dnorm (l:ℝ→ℝ) := ∑' a:MI m,q.Dterm l a |
| |
| def RR (x:Rn m) := q.Amat x-q.Lmat x |
| def sigterm (l:ℝ→ℝ) (a:MI m) := Pj q.covMat (cof q.RR a) (q.DD l a) |
| def sigma (l:ℝ→ℝ) := 2 * ∑' a,q.sigterm l a |
|
|
| lemma H_K_split {l} (hl:TestF l) : |
| let f := q.Hmat (fl l) |
| let b := q.FL.eval (pl l) |
| let W := q.covMat |
| HasSum (q.Dterm l) (KP W f f-2*Pt W f b+Pt W b b+q.dirF W l) := by |
| intro f b W |
| have hf : regular f:= q.regH (fl_test hl) |
| have hb : regular b:=q.FL.eval_reg (pl_test hl) |
| have h₁ := (K_sum (B:=W) hf hf).hasSum |
| have h₂ := pt_sum (B:=W) hf hb |
| have h₃ := pt_sum (B:=W) hb hb |
| have h₄ := q.Pt_dir W hl |
| have he (a:MI m) : |
| q.Dterm l a=CpPair W f f a*kMode a-2*CpPair W f b a+ |
| CpPair W b b a+(deg a:ℝ)*CpPair W b b a := by |
| change (1+_)*Pj W (kMode a • cof f a - cof b a) (kMode a • cof f a - cof b a)=_ |
| simp only [pj_ss,pJ_smul_left,pJ_smul_right] |
| rw [pj_sym W (cof b _)] |
| unfold CpPair kMode |
| have hi : 0 < 1+(deg a:ℝ) := by positivity |
| field_simp; ring |
| rw [funext he] |
| exact (((h₁.sub (h₂.mul_left 2)).add h₃).add h₄) |
|
|
| lemma R_reg : regular q.RR := |
| ⟨q.A_poly.sub q.L_poly,q.A_SM.aestronglyMeasurable.sub q.L_cont.aestronglyMeasurable⟩ |
| lemma sig_split {l:ℝ→ℝ} (hl:TestF l) : |
| let f := q.Hmat (fl l) |
| let b := q.FL.eval (pl l) |
| let W := q.covMat |
| HasSum (q.sigterm l) (KP W q.RR f-Pt W q.RR b) := by |
| intro f b W |
| have hf : regular f:= q.regH (fl_test hl) |
| have hb : regular b:=q.FL.eval_reg (pl_test hl) |
| have h₁ := (K_sum (B:=W) q.R_reg hf).hasSum |
| have h₂ := pt_sum (B:=W) q.R_reg hb |
| have he (a:MI m) : q.sigterm l a=CpPair W q.RR f a*kMode a-CpPair W q.RR b a := by |
| unfold sigterm DD CpPair |
| have hi (A B C:Mat m) : Pj W A (B-C)= Pj W A B-Pj W A C := by |
| simp_rw [← pJ_eq]; rw [_root_.map_sub] |
| rw [show q.covMat=W from rfl,hi,pJ_smul_right,mul_comm] |
| rw [funext he] |
| exact h₁.sub h₂ |
|
|
| def Jtest (W:Mat m) (l:ℝ→ℝ) := |
| etw W (fun x=> q.Lmat x*q.FL.eval (pl (fun z=>l z^2)) x)-q.dirF W l |
| end ProjField |
| end GeneralMahler |
|
|
| end |
|
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| end OAI |
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