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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | 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
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