Definition lebint_s
(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''':
sigalg_2 _ F)(u:forall A,F A->R)(u':msur_0 _ _ F' F'' F''' u)(u'':msur_1 _ _ F'
F'' F''' u)(u''':msur_2 _ _ F' F'' F''' u)(u'''':msur_3 _ _ F' F'' F''' u)(n:
nat)(v:nat->R)(w:nat->Ensemble U)(S0:simple_0 _ n v w)(S1:simple_1 _ n v w)(S2:
simple_2 _ n v w)(S3:simple_3 _ n v w)(M:msurable _ _ F' F'' F'''(fun x:U=>v(
proj1_sig(simple_P0 _ n v w S0 S1 S2 S3 x))))(E:Ensemble U)(E':F E)
:=sum_f_R0 (fun i:nat=>((v i)*(u(Intersection _(w i)E)(sigalg_P2 _ _ F' F''
F'''(w i)E(simple_P1 _ _ F' F'' F''' n v w S0 S1 S2 S3 M i)E')))%R) n.Definition lebint
(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''':
sigalg_2 _ F)(u:forall A,F A->R)(u':msur_0 _ _ F' F'' F''' u)(u'':msur_1 _ _ F'
F'' F''' u)(u''':msur_2 _ _ F' F'' F''' u)(u'''':msur_3 _ _ F' F'' F''' u)(f:U
->R)(f':forall x:U,(f x>=0)%R)(M:msurable _ _ F' F'' F''' f)(E:Ensemble U)(E':F
E)
:=let B:=fun l=>forall n,forall v,forall w,forall S0,forall S1,forall S2,forall
S3,forall M',(forall x:U,(0<=(fun y:U=>v(proj1_sig(simple_P0 _ n v w S0 S1 S2
S3 y)))x<=f x)%R)->(lebint_s _ _ _ _ _ u u' u'' u''' u'''' n v w S0 S1 S2 S3 M'
E E'<=l)%R in fun L=>B L/\forall L',B L'->(L'<=L)%R.

(* Specifications: *)
Lemma lebint_P0: (forall(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''' :sigalg_2 _ F)(u:forall A,F A->R)(u':msur_0 _ _ F' F'' F''' u)(u'':msur_1 _ _ F' F'' F''' u)(u''':msur_2 _ _ F' F'' F''' u)(u'''':msur_3 _ _ F' F'' F''' u)(n :nat)(v:nat->R)(w:nat->Ensemble U)(S0:simple_0 _ n v w)(S1:simple_1 _ n v w)(S2 :simple_2 _ n v w)(S3:simple_3 _ n v w)(M:msurable _ _ F' F'' F'''(fun x:U=>v( proj1_sig(simple_P0 _ n v w S0 S1 S2 S3 x))))(g:U->R)(g':forall x:U,g x>=0)(g'' :msurable _ _ F' F'' F''' g)(E:Ensemble U)(E':F E), (forall y:U,(fun x:U=>v(proj1_sig(simple_P0 _ n v w S0 S1 S2 S3 x)))y<=g y)-> forall L,lebint _ _ _ _ _ u u' u'' u''' u'''' g g' g'' E E' L->lebint_s _ _ F' F'' F''' u u' u'' u''' u'''' n v w S0 S1 S2 S3 M E E'<=L)%R.
Lemma u_cmp: forall(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''': sigalg_2 _ F)(u:forall A,F A->R)(u'''':msur_3 _ _ F' F'' F''' u)A B A' B', A=B->u A A'=u B B'.
Lemma lebint_P1: forall(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''': sigalg_2 _ F)(u:forall A,F A->R)(u':msur_0 _ _ F' F'' F''' u)(u'':msur_1 _ _ F' F'' F''' u)(u''':msur_2 _ _ F' F'' F''' u)(u'''':msur_3 _ _ F' F'' F''' u)(f:U ->R)(f':forall x:U,(f x>=0)%R)(f'':msurable _ _ F' F'' F''' f)(E:Ensemble U)(E' :F E), forall(A:Ensemble U)(A':F A)(A'':Included _ A E), lebint_s _ _ F' F'' F''' u u' u'' u''' u'''' 2(fun i:nat=>INR i)(fun i:nat=>if eq_nat_dec i 0 then Complement _ A else if eq_nat_dec i 1 then A else Empty_set _)(simple_P4 _ _)(simple_P5 _ _)(simple_P6 _ _)(simple_P7 _ _)(simple_P8 _ _ F' F'' F''' A A')E E'=u A A'.
Lemma misc_P0 (U:Type)(f:U->R)(f':forall x:U,(f x>=0)%R) :forall a:R,(0<a)%R->forall x:U,((fun y:U=>(f y*a))x>=0)%R.
Lemma lebint_P2: (forall(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''' :sigalg_2 _ F)(u:forall A,F A->R)(u':msur_0 _ _ F' F'' F''' u)(u'':msur_1 _ _ F' F'' F''' u)(u''':msur_2 _ _ F' F'' F''' u)(u'''':msur_3 _ _ F' F'' F''' u)(n :nat)(v:nat->R)(w:nat->Ensemble U)(S0:simple_0 _ n v w)(S1:simple_1 _ n v w)(S2 :simple_2 _ n v w)(S3:simple_3 _ n v w)(M:msurable _ _ F' F'' F'''(fun x:U=>v( proj1_sig(simple_P0 _ n v w S0 S1 S2 S3 x))))(E:Ensemble U)(E':F E), forall(c:R)(c':0<c),lebint_s _ _ _ _ _ u u' u'' u''' u'''' n(fun i:nat=>v i*c)w (simple_P9 _ n v w S0 S1 S2 S3 c c')(simple_P10 _ n v w S0 S1 S2 S3 c c')( simple_P11 _ n v w S0 S1 S2 S3 c c')(simple_P12 _ n v w S0 S1 S2 S3 c c')( simple_P13 _ _ F' F'' F''' n v w S0 S1 S2 S3 M c c') E E' = c*lebint_s _ _ _ _ _ u u' u'' u''' u'''' n v w S0 S1 S2 S3 M E E')%R.
Theorem markov: (forall(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''' :sigalg_2 _ F)(u:forall A,F A->R)(u':msur_0 _ _ F' F'' F''' u)(u'':msur_1 _ _ F' F'' F''' u)(u''':msur_2 _ _ F' F'' F''' u)(u'''':msur_3 _ _ F' F'' F''' u)(f :U->R)(f':forall x:U,f x>=0)(f'':msurable _ _ F' F'' F''' f)(a:R)(a':0<a), forall L,lebint _ _ _ _ _ u u' u'' u''' u'''' f f' f''(Full_set U)(sigalg_P3 U F F' F'' F''') L ->u(fun x:U=>f x>=a)(msurable_P1 _ _ F' F'' F''' f f' f'' a a')<=/a*L)%R.