Quelle power_series_integ.pvs Sprache: PVS

power_series_integ[T: TYPE+ from real]: THEORY
%----------------------------------------------------------------------------
%
% Term by term integration of power series:
%
%                         n
%                     ----
%  series(integ_powerseq(a, x)) =   \     a(k)*x^(k+1)/k+1
%                     /
%                     ----
%                     k = 0
%
%
% The intention here is that one passes in the domain of convergence [T]
% of the power series.  This will either be all of the reals or {x| -R < x < R}
% where R is the range of convergence.  Most of the lemmas assume
% convergence of the power series a: conv_powerseries?(a) =
%     (FORALL (x: T): convergent?(powerseries(a)(x)))
%
%
% Author: Ricky W. Butler        10/2/04
%
% NOTES
%  * Prelude restrictions on ^ [i.e. ^(r: real, i:{i:int | r /= 0 OR i >= 0}]
%    lead to IF-THEN-ELSE on definition of "integ_powerseq"
%  * The alternate form: integseq(a): (LAMBDA (k: nat): (k+1)*a(k+1))avoids
%    the difficulty by shifting (i.e. starting with 2nd term)
%  * These are shown to be equivalent in lemma "eriv_series_shift"
%
%  Author: Ricky W. Butler                NASA Langley Research Center
%----------------------------------------------------------------------------
BEGIN

   ASSUMING  %% T is either "real" or a open ball of radius R about 0
     IMPORTING analysis@deriv_domain_def

    connected_domain : ASSUMPTION connected?[T]

    not_one_element  : ASSUMPTION not_one_element?[T]

    open            : ASSUMPTION
         FORALL (x : T) : EXISTS (delta : posreal): FORALL (y: real):
                            abs(x-y) < delta IMPLIES T_pred(y)

    ball: ASSUMPTION FORALL (x: T): T_pred(x) IMPLIES T_pred(-x)

   ENDASSUMING


   x,xp, x0: VAR T
   k,n: VAR nat
   a: VAR sequence[real]
   t: VAR real

   epsilon: VAR posreal

%  Integative of a infinite power series is the term-by-term integative

   IMPORTING analysis@deriv_domain_def

   deriv_domain: LEMMA deriv_domain?[T]

   IMPORTING power_series_conv[T], analysis@derivatives, analysis@taylors

   integseq(a): sequence[real] = (LAMBDA (k: nat): IF k = 0 THEN 0
                                                   ELSE a(k-1)/k ENDIF)

   integ_powerseq(a,x): sequence[real] = (LAMBDA k:  a(k)*x^(k+1)/(k+1))


%  -------- term by term integration is  convergent ------------------------

   conv_integseq: LEMMA conv_powerseries?(a) IMPLIES % Salas Hille page 454
                           conv_powerseries?[T](integseq(a))

   integ_powerseries_conv: THEOREM
                          conv_powerseries?(a) IMPLIES
              (FORALL (x: T): convergent?(series(integ_powerseq(a, x))))

   integseq_lim_shift: LEMMA conv_powerseries?(a) IMPLIES
                         (LAMBDA x: limit(series(powerseq(integseq(a), x)))) =
                         (LAMBDA x: limit(series(integ_powerseq(a, x))))

   IMPORTING  power_series_deriv[T],
              analysis@integral[T],
              analysis@fundamental_theorem[T],
              analysis@indefinite_integral[T]

   powerseries_integrable?: LEMMA conv_powerseries?(a) IMPLIES
                             integrable?[T](LAMBDA x: limit(powerseries(a)(x)))

   powerseries_integral: THEOREM   conv_powerseries?(a) IMPLIES
       (
          LET f = (LAMBDA x: limit(powerseries(a)(x))) IN
          LET g = (LAMBDA x: limit(series(integ_powerseq(a,x)))) IN
                   EXISTS (C: real): integral[T](f) = g + const_fun(C) )

   integral_powerseries: THEOREM   T_pred(0) AND conv_powerseries?(a) IMPLIES
          LET f = (LAMBDA x: limit(powerseries(a)(x))) IN
              Integrable?(0,x,f) AND
              Integral(0,x,f) = limit(series(integ_powerseq(a,x)))

END power_series_integ

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