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Datei: integral_bounded.pvs Sprache: PVS

Original von: PVS^©

integral_bounded[T: TYPE FROM real]: THEORY
%------------------------------------------------------------------------------
%
%  An Integrable Function is bounded
%
%  Author:  Rick Butler               NASA Langley
%
%------------------------------------------------------------------------------
BEGIN

   ASSUMING
      IMPORTING deriv_domain_def[T]

      connected_domain : ASSUMPTION connected?[T]

      not_one_element : ASSUMPTION not_one_element?[T]

   ENDASSUMING

   AUTO_REWRITE+ not_one_element

   IMPORTING reals@sigma_below,
             integral_prep[T]

   a,b,c,d,x,y,z: VAR T

   f,f1,f2,g: VAR [T -> real]

   eps, delta: VAR posreal

   xv,yv: VAR real


%    integrable?(a:T,b:{x:T|a<x})(f:[T->real]): bool =
%                     integrable?(a,b,f)

   bounded_on?(a,b,f): bool = (EXISTS (B: real):
                   (FORALL (x: closed_interval(a,b)): abs(f(x)) <= B))

   int_to_bnd: LEMMA  % Rosenlicht pg 122
                    a < b IMPLIES
               (integrable?(a,b,f) IMPLIES
                   (EXISTS (EP: partition[T](a,b)):
                           FORALL (j: below(length(EP) - 1)):
                              bounded_on?(EP(j), EP(1 + j), f)))

   bounded_on_all?(a:T,b:{x:T|a<x},P: partition[T](a,b))(f:[T->real]): bool =
             (FORALL (j: below(P`length - 1)): bounded_on?((P(j),P(j+1),f)))


   bounded_on_all_lem: LEMMA  % Rosenlicht pg 122
                    a < b IMPLIES
               integrable?(a,b,f) IMPLIES
                   (EXISTS (P: partition[T](a,b)): bounded_on_all?(a,b,P)(f))

   MINj_prep: LEMMA FORALL (a: T, b: {x: T | a < x},
                            P: partition[T](a, b),
                            f: (bounded_on_all?(a, b, P)),
                            j: below(P`length - 1)):
     nonempty?[real] ({fx: real | EXISTS (xx: T):
           P`seq(j) <= xx AND xx <= P`seq(1 + j) AND fx = f(xx)})
      AND
      bounded?({fx: real | EXISTS (xx: T):
   P`seq(j) <= xx AND xx <= P`seq(1 + j) AND fx = f(xx)})

   MINj(a:T,b:{x:T|a<x},P: partition[T](a,b),j: below(length(P)-1),
             f: (bounded_on_all?(a,b,P))): real =
                  glb({fx: real | EXISTS (xx: T):
                                  P`seq(j) <= xx AND xx <= P`seq(1 + j) AND
                                  fx = f(xx)})

   MINj_lem: LEMMA a < b IMPLIES
                     FORALL (P: partition[T](a, b),
                             f: (bounded_on_all?(a, b, P)),
                             j: below(length(P)-1),
                             x: closed_interval[T](seq(P)(j), seq(P)(1+j))):
                         MINj(a,b,P,j,f) <= f(x)

   MAXj(a:T,b:{x:T|a<x},P: partition[T](a,b),j: below(length(P)-1),
            f: (bounded_on_all?(a,b,P))): real =
                  lub({fx: real | EXISTS (xx: T):
                                  P`seq(j) <= xx AND xx <= P`seq(1 + j) AND
                                  fx = f(xx)})

   MAXj_lem: LEMMA a < b IMPLIES
                     FORALL (P: partition[T](a, b),
                             f: (bounded_on_all?(a, b, P)),
                             j: below(length(P)-1),
                             x: closed_interval[T](seq(P)(j), seq(P)(1+j))):
                         MAXj(a,b,P,j,f) >= f(x)

    MIN_ALL(a:T,b:{x:T|a<x},P: partition[T](a,b),
            f: (bounded_on_all?(a,b,P))): real =
         min({mm: real | EXISTS (jj: below(length(P) - 1)):
                              mm = MINj(a,b,P,jj,f)})

    MAX_ALL(a:T,b:{x:T|a<x},P: partition[T](a,b),
           f: (bounded_on_all?(a,b,P))): real =
           max({mm: real | EXISTS (jj: below(length(P) - 1)):
                             mm = MAXj(a,b,P,jj,f)})

    MIN_ALL_lem: LEMMA  a < b IMPLIES
                    FORALL (P: partition[T](a, b),
                            f: (bounded_on_all?(a, b, P)),
                            x: closed_interval(a,b)):
                         MIN_ALL(a, b, P, f) <= f(x)

    MAX_ALL_lem: LEMMA  a < b IMPLIES
                    FORALL (P: partition[T](a, b),
                            f: (bounded_on_all?(a, b, P)),
                            x: closed_interval(a,b)):
                         MAX_ALL(a, b, P, f) >= f(x)

   bounded_on_all_is: LEMMA a < b AND
                           (EXISTS (P: partition[T](a,b)):
                                   bounded_on_all?(a,b,P)(f))
                           IMPLIES bounded_on?(a, b, f)

   integrable_bounded: LEMMA a < b AND % Rosenlicht pg 122
                             integrable?(a,b,f)
                          IMPLIES bounded_on?(a, b, f)

   bnded_on?(a:T,b:{x:T|a<x})(f:[T->real]): bool = bounded_on?(a, b, f)

   bnd_on_lem: LEMMA a < b IMPLIES FORALL (P: partition[T](a,b)):
                     bounded_on?(a, b, f)
                  IMPLIES  bounded_on_all?(a, b, P)(f)

   integrable_bounded_on_all: LEMMA a < b IMPLIES
                  FORALL (P: partition[T](a,b)):
                  integrable?(a, b, f) IMPLIES
                         bounded_on_all?(a, b, P)(f)

END integral_bounded

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