| import OAI.Geometry.Convex.GeneralMahler.ScalarForm |
| import OAI.Geometry.Convex.GeneralMahler.Eq13 |
|
|
| namespace OAI |
| /-! §04 profile quadratic identity. -/ |
| noncomputable section |
| open Set Filter MeasureTheory MeasureTheory.Measure Matrix Real Metric |
| open scoped Topology NNReal ENNReal MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace Interval |
| namespace GeneralMahler |
| open HMode Profile Layers |
| variable {m:ℕ} [NeZero m] |
|
|
| namespace ProjField |
| omit [NeZero m] in |
| lemma tw_add {B:FieldMat m} {f j:ℝ→ℝ} (W:Mat m) (hf:TestF f) (hj:TestF j) : |
| LDel B W (fun x=> f x+j x)=LDel B W f+LDel B W j := by |
| have h : (fun x=> f x) = fun x=> (f x+j x)-j x := by ext; ring |
| have he := Delta_sub B W (hf.add hj) hj |
| rw [← h] at he; linarith |
| |
| variable (q:ProjField m) |
|
|
| lemma evId (x:Rn m) : q.FL.eval (fun x=> x) x=q.Lmat x := by |
| unfold FieldMat.eval |
| rw [cfc_id' ℝ _ (show IsSelfAdjoint (q.FL.A x) from q.FL.sym x)]; rfl |
| |
| lemma Pb_sq {l} (hl:TestF l) (W:Mat m) : |
| Pt W (q.FL.eval l) (q.FL.eval l)=etw W (q.FL.eval (fun x=>l x*l x)) := by |
| unfold Pt Pj etw et |
| have he (A:Mat m): jprod A A=A*A := by unfold jprod; module |
| simp_rw [he,FieldMat.eval_mul _ hl hl] |
| |
| lemma form_sq {l} (hl:TestF l) : |
| q.Dnorm l+q.Jtest q.covMat l = |
| q.delt (fun z=>l z^2)-LDel q.FL q.covMat (ps (fun z=> l z^2))+ |
| 2*q.be q.FL q.covMat (fl l) (pl l)+q.Ct (fl l) (fl l) := by |
| let f := fl l |
| let b := pl l |
| let h := fun x=>l x^2 |
| let W := q.covMat |
| have hf : TestF f := fl_test hl |
| have hb : TestF b := pl_test hl |
| have ht : TestF h := by simp_rw [h,pow_two]; exact hl.mul hl |
| let w := prim (fun x=>f x*deriv b x) |
| let v := prim (fun x=>deriv f x*b x+deriv f x*b x) |
| have hw := p_test (hf.mul hb.der) |
| have hv := p_test ((hf.der.mul hb).add (hf.der.mul hb)) |
| have dw (x) : deriv w x=f x*deriv b x := (prim_d (hf.mul hb.der) x).deriv |
| have dv (x) : deriv v x=deriv (Nr b) x*b x+deriv (Nr b) x*b x := |
| (prim_d ((hf.der.mul hb).add (hf.der.mul hb)) x).deriv |
| let F := fun x=>b x*b x+x*pl h x |
| have Ht := pl_test ht |
| have HI := TestF.id.mul Ht |
| have HB := hb.mul hb |
| have FF : TestF F := HB.add HI |
| let L := LDel q.FL W |
|
|
| have hd1 : q.Idw (fun x=>v x-2*(f x*b x)) = q.Idw (fun x=> -(2*w x)) := by |
| apply Id_ext _ (hv.sub ((TestF.const _).mul (hf.mul hb))) (((TestF.const _).mul hw).neg) |
| funext x |
| have hx := ((hf.diff x).hasDerivAt.fun_mul (hb.diff x).hasDerivAt).const_mul 2 |
| have he := (((hv.diff x).hasDerivAt).fun_sub hx).deriv |
| have hh : deriv (fun x=> -(2*w x)) x= -(2*deriv w x) := |
| (((hw.diff x).hasDerivAt.const_mul 2).neg).deriv |
| rw [he,hh,dv,dw] |
| change _+_-_=_; dsimp only [f,fl,b]; ring |
| have IdH : q.Idw v-2*q.Idw (fun x=>f x*b x) = - (2*q.Idw w) := by |
| have ha : (fun x=> -(2*w x))=fun x=> -2*w x := by ext; ring |
| rw [q.Id_sub hv ((TestF.const _).mul (hf.mul hb)), q.Id_cmul, ha, q.Id_cmul] at hd1 |
| linarith |
| |
| have hd2 : L (fun x=>F x-2*w x)= L (fun x=> - (ps h x)) := by |
| apply L_ext _ _ (FF.sub ((TestF.const _).mul hw)) (ps_test ht).neg |
| funext x |
| have he := (((((hb.diff x).hasDerivAt).fun_mul (hb.diff x).hasDerivAt).fun_add |
| ((hasDerivAt_id' x).fun_mul (Ht.diff x).hasDerivAt)).fun_sub |
| ((hw.diff x).hasDerivAt.const_mul 2)).deriv |
| have HH : deriv (fun x=> -ps h x) x= -deriv (ps h) x := |
| (((ps_test ht).diff x).hasDerivAt.neg).deriv |
| change deriv (fun x=>F x-2*w x) x=_ at he |
| rw [he,HH,dw,ps_d ht x] |
| unfold f fl Nr N b |
| simp only [pl_d ht,pl_d hl] |
| have hi : deriv h x= deriv l x*l x + l x*deriv l x := by |
| unfold h; simp_rw [pow_two] |
| exact ((hl.diff x).hasDerivAt.fun_mul (hl.diff x).hasDerivAt).deriv |
| rw [hi] |
| unfold h; ring |
| have LdH : L F-2*L w= -L (ps h) := by |
| have he : (fun x=> -ps h x) = fun x=> (-1)*ps h x := by ext; ring |
| unfold L at * |
| rw [Delta_sub _ _ FF ((TestF.const _).mul hw),L_cmul _ _ _ hw,he, |
| L_cmul _ _ _ (ps_test ht)] at hd2 |
| linarith |
| |
| have hgas : ga F=ga (fun x=> f x*b x) := by |
| have hh := ga_parts Ht |
| rw [pl_d ht] at hh |
| unfold F; rw [ga_add HB HI,hh, show f=Nr b from rfl, |
| ga_nr_parts hb hb,show deriv b=l from pl_d hl] |
| simp_rw [h,pow_two] |
| |
| have heq := q.eq14 hb hb hv dv |
| rw [rightLayer.Q0_cov hb (nr_test hb)] at heq |
| have hu : (fun x=>deriv b x*deriv b x)=h := by simp only [b,pl_d hl,h,pow_two] |
| rw [hu] at heq |
| change q.Lr.Q f f-q.s0*ga (fun x=>f x*b x)=q.Idw v+q.delt h at heq |
| have hi : Pt W (q.FL.eval b) (q.FL.eval b)+ |
| etw W (fun x=> q.Lmat x*q.FL.eval (pl h) x) = L F + q.s0*ga F := by |
| have he (x) : q.Lmat x*q.FL.eval (pl h) x=q.FL.eval (fun x=> x*pl h x) x := by |
| rw [FieldMat.eval_mul q.FL TestF.id Ht,q.evId] |
| rw [Pb_sq q hb]; simp_rw [he]; rw [L_formula _ W HB,L_formula _ W HI] |
| unfold L F; rw [tw_add W HB HI, ga_add HB HI]; change _=_ |
| rw [show trN W=q.s0 from rfl]; ring |
| have hg := fl_ga hl |
| have hcov := q.eq13 hf hf hg hg |
| have hpa := q.f_pair16 q.FL hf hb hg hw dw |
| have hk := q.dm_e (hf.mul hb) |
| have h1 := (q.H_K_split hl).tsum_eq |
| change q.Dnorm l= _ at h1 |
| unfold Jtest |
| change q.Dnorm _+ (_-q.dirF W l) = q.delt h-L (ps h)+2*q.be q.FL W f b+q.Ct f f |
| change Pt W _ _=q.Iw (fun x=>f x*b x)+(L w-q.Idw w)-q.be q.FL W f b at hpa |
| rw [hgas] at hi; rw [h1]; dsimp only [W,h,f,b] at *; linarith |
| end ProjField |
| end GeneralMahler |
| |
| end |
| |
| end OAI |
| |
