| import OAI.Geometry.Convex.GeneralMahler.LowDegree |
|
|
| namespace OAI |
| /-! Linear profile identity, first part of Eq17. -/ |
| 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 Profile Layers HMode |
| variable {m:ℕ} [NeZero m] |
|
|
| namespace ProjField |
| variable (q:ProjField m) |
| theorem form_lin {l:ℝ→ℝ} (hl:TestF l) : |
| q.delt l=LDel q.FL q.covMat (ps l)+ q.sigma l- |
| 2*q.be q.FL q.covMat (fun x=>x) (pl l) -q.be q.FL q.covMat (fl l) (fun x=>x) |
| -2*q.Ct (fun x=>x) (fl l) := by |
| let i := fun x:ℝ=>x |
| let k := fun x:ℝ=>(1/2:ℝ)*x |
| let b := pl l |
| let f := fl l |
| let W := q.covMat |
| let A := LDel q.FL W |
| have hf : TestF f := fl_test hl |
| have hb : TestF b := pl_test hl |
| have ht := ps_test hl |
| have hi : TestF i := TestF.id |
| have hk : TestF k := (TestF.const _).mul hi |
| have dk : deriv k=fun _=>1/2 := by |
| ext x; unfold k; convert (((hasDerivAt_id' x).const_mul (1/2:ℝ)).deriv) using 1 ; ring |
| have di : deriv i=fun _=>1 := by ext; simp [i] |
| have K : Nr k=i := by |
| unfold Nr N; rw [dk]; ext x; unfold k i; simp; ring |
| let w := prim f |
| let v := prim (fun x=>b x+deriv f x*k x) |
| let u := prim b |
| have hw := p_test hf |
| have hv := p_test (hb.add (hf.der.mul hk)) |
| have hu := p_test hb |
| have du : deriv u=b := funext fun x=> (prim_d hb x).deriv |
| have dw : deriv w=f := funext fun x=> (prim_d hf x).deriv |
| have dv (x:ℝ) : deriv v x=deriv (Nr k) x*b x+deriv (Nr b) x*k x := by |
| rw [K,di] |
| have he := (prim_d (hb.add (hf.der.mul hk)) x).deriv |
| simp only [one_mul]; exact he |
|
|
| let I := fun x=> f x*x |
| have ih := hf.mul hi |
| have h₁ : q.Iw I-2*q.Lr.Q i f=q.Idw (ps l)-q.delt l := by |
| have hh := q.eq14 hk hb hv dv |
| rw [K,rightLayer.Q0_cov hb hi] at hh |
| change q.Lr.Q i f-q.s0*ga (fun x=>i x*b x)=q.Idw v+q.delt _ at hh |
| have hz : (fun x=>deriv k x*deriv b x) =fun x=>(1/2:ℝ)*l x := by rw [dk,show deriv b=l from pl_d hl] |
| rw [hz,q.delt_scale _ hl] at hh |
| have hd := q.dm_e ih |
| have he : ga I=2*ga (fun x=>i x*b x) := by |
| have HH : ga (fun x=>deriv b x*deriv i x) = ga (deriv b) := by rw [di]; simp |
| change ga (fun x=>Nr b x*i x)=_ |
| rw [ga_nr_parts hb hi,HH, ← ga_parts hb] |
| rw [ga_prod_comm b i] |
| change ga _+ga (fun x=>i x*b x)=_ |
| ring |
| have ht' : q.Idw (fun x=> I x-2*v x)=q.Idw (ps l) := by |
| apply q.Id_ext (ih.sub ((TestF.const _).mul hv)) ht |
| ext x |
| have hd := ((((hf.diff x).hasDerivAt.fun_mul (hi.diff x).hasDerivAt)).fun_sub |
| ((hv.diff x).hasDerivAt.const_mul 2)).deriv |
| change deriv (fun x=>f x*i x-2*v x) x=_ |
| rw [hd,dv,K,ps_d hl,di] |
| unfold f b i k fl; ring |
| rw [q.Id_sub ih ((TestF.const _).mul hv),q.Id_cmul] at ht' |
| change q.Idw I=q.Iw I-q.s0*ga I at hd |
| rw [he] at hd; linarith |
| have hh : deriv (fun x=>w x-ps l x)=deriv (fun x=>2*u x) := by |
| ext x |
| rw [((hw.diff x).hasDerivAt.fun_sub (ht.diff x).hasDerivAt).deriv, |
| ((hu.diff x).hasDerivAt.const_mul 2).deriv,du,dw,ps_d hl] |
| unfold b f; ring |
| have he := L_ext q.FL W (hw.sub ht) ((TestF.const _).mul hu) hh |
| rw [Delta_sub _ W hw ht,L_cmul q.FL W _ hu] at he |
| have hx := q.Id_ext (hw.sub ht) ((TestF.const _).mul hu) hh |
| rw [q.Id_sub hw ht,q.Id_cmul] at hx |
| have hvu := q.h_weighted hu |
| change etw W (q.Hmat u) = q.Idw u at hvu |
| rw [q.etw16a q.FL W hu,du,q.MB_id] at hvu |
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|
| have hcov := q.eq13 hi hf Profile.id_ga (fl_ga hl) |
| change KP W q.Amat (q.Hmat f)=_ at hcov |
| have ho : ∀ x,deriv w x=f x*deriv i x := by rw [di,dw]; simp |
| have hpa := q.f_pair16 (B:=q.FL) hf hi (fl_ga hl) hw ho |
| rw [show q.FL.eval i=q.Lmat from funext q.evId,pt_sym] at hpa |
| have hk := q.k_l W (q.regH hf) |
| have hq := q.kpR W (q.regH hf) |
| have hy := (q.sig_split hl).tsum_eq |
| change Pt W q.Lmat (q.Hmat f)=q.Iw I+(A w-q.Idw w)-q.be q.FL W f i at hpa |
| unfold sigma; rw [hy] |
| linarith |
| end ProjField |
| end GeneralMahler |
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| end |
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| end OAI |
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