File size: 4,022 Bytes
8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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
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
end
end OAI
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