Quelle integral_def.pvs Sprache: PVS

integral_def[T: TYPE FROM real]: THEORY
%----------------------------------------------------------------------------
%
%    Definition of Riemann Integral
%    Author: Rick Butler
%    Major Revision:  12/1/03
%
%----------------------------------------------------------------------------
BEGIN

   ASSUMING
      IMPORTING deriv_domain_def[T]

      connected_domain : ASSUMPTION connected?[T]

      not_one_element : ASSUMPTION not_one_element?[T]

   ENDASSUMING

   IMPORTING finite_sets@finite_sets_minmax[real,<=] %, finite_sets_more

   AUTO_REWRITE+ finseq_appl

   a,b,x: VAR T
   f,g: VAR [T -> real]

%   open_interval  (a:T,b:{x:T|a<x}): TYPE = { x | a <  x AND x <  b}
%   closed_interval(a:T,b:{x:T|a<x}): TYPE = { x | a <= x AND x <= b}

   IMPORTING reals@intervals_real[T]

% % Unlike Rosenlicht index runs from 0 to N-1:

%    partition(a:T,b:{x:T|a<x}): TYPE =
%               {fs: finite_sequence[closed_interval(a,b)] |
%                     Let N = length(fs), xx = seq(fs) IN
%                     N > 1 AND xx(0) = a AND xx(N-1) = b AND
%                     (FORALL (ii: below(N-1)): xx(ii) < xx(ii+1))}

%    width(a:T, b:{x:T|a<x}, P: partition(a,b)): posreal =
%              LET xx = seq(P), N = length(P) IN
%                       max({ l: real | EXISTS (ii: below(N-1)):
%                                          l = xx(ii+1) - xx(ii)})

%    width_lem     : LEMMA (FORALL (a:T), (b:{x:T|a<x}),
%                             (P: partition(a,b)),(ii: below(length(P)-1)):
%                          width(a,b,P) >= P(ii+1) - P(ii))

%    parts_order   : LEMMA  FORALL (P: partition(a,b), ii,jj: below(length(P))):
%                               ii < jj IMPLIES seq(P)(ii) < seq(P)(jj)

%    parts_order_le: LEMMA FORALL (P: partition(a,b),ii,jj: below(length(P))):
%                              ii <= jj IMPLIES seq(P)(ii) <= seq(P)(jj)


%    parts_disjoint: LEMMA FORALL (P: partition(a,b), ii,jj: below(length(P)-1)):
%                              seq(P)(ii) < x AND x < seq(P)(1 + ii) AND
%                              seq(P)(jj) < x AND x < seq(P)(1 + jj)
%                              IMPLIES
%                                 jj = ii

%    in_part?(a:T, b:{x:T|a<x},P: partition(a,b),xx:T): MACRO bool =
%                         EXISTS (ii: below(length(P)-1)): xx = seq(P)(ii)

%    in_sect?(a:T, b:{x:T|a<x},P: partition(a,b),
%             ii: below(length(P)-1),xx:T): MACRO bool =
%                       seq(P)(ii) < xx AND xx < seq(P)(ii+1)

%    part_in       : LEMMA FORALL (P: partition(a,b)):
%                              a < b AND a <= x AND x <= b IMPLIES
%                                  (EXISTS (ii: below(length(P)-1)):
%                                        seq(P)(ii) <= x AND x <= seq(P)(ii+1))

%    part_not_in       : LEMMA a < b IMPLIES FORALL (P: partition(a,b)):
%                                 FORALL (ii,jj: below(length(P)-1)):
%                                   seq(P)(ii) < x AND x < seq(P)(ii+1)
%                                 IMPLIES x /= seq(P)(jj)

%    Prop: VAR [T -> bool]
%    part_induction: LEMMA  (FORALL (P: partition(a,b)):
%                               (FORALL ( x: closed_interval(a,b)):
%                            LET xx = seq(P), N = length(P) IN
%                    (FORALL (ii : below(N-1)):
%                            xx(ii) <= x AND x <= xx(ii+1) IMPLIES
%                                 Prop(x))
%                   IMPLIES Prop(x) ) )

%    IMPORTING reals@sigma_below, reals@sigma_upto

%    eq_partition(a:T,b:{x:T|a<x},N: above(1)): partition(a,b) =
%                   (# length := N,
%                      seq := (LAMBDA (ii: below(N)): a + ii*(b-a)/(N-1)) #)

%    xis?(a:T,b:{x:T|a<x},P:partition(a,b))
%          (fs: [below(length(P)-1) -> closed_interval(a,b)]): bool =
%               (FORALL (ii: below(length(P)-1)):
%                      P(ii) <= fs(ii) AND fs(ii) <= P(ii+1))

%    AUTO_REWRITE+ xis?

%    xis_lem: LEMMA FORALL (P: partition(a,b), (xis: (xis?(a,b,P))),
%                          (ii: below(length(P)-1))):
%                          P(ii) <= xis(ii) AND xis(ii) <= P(ii+1)

%    AUTO_REWRITE+ xis_lem

%    Rie_sum(a:T,b:{x:T|a<x},P:partition(a,b),
%                   xis: (xis?(a,b,P)),f:[T->real]): real =
%                   LET N = length(P)-1 IN
%           sigma[below(N)](0,N-1,(LAMBDA (n: below(N)):
%                                        (P(n+1) - P(n)) * f(xis(n))))

%    Rie_sec(a:T,b:{x:T|a<x},P:partition(a,b), xis: (xis?(a,b,P)), f:[T->real],
%            n: upto(length(P)-1)): real =
%                            IF n = 0 THEN 0
%                            ELSE (seq(P)(n) - seq(P)(n - 1)) * f(xis(n-1))
%                            ENDIF

%    Rie_sum_alt(a:T,b:{x:T|a<x},P:partition(a,b),
%                   xis: (xis?(a,b,P)),f:[T->real]): real =
%                   LET N = length(P)-1 IN
%                      sigma[upto(N)](1,N,(LAMBDA (n: upto(N)):
%                             Rie_sec(a,b,P,xis,f,n)))

%    Rie_sum_alt_lem: LEMMA a < b IMPLIES
%                             FORALL (P: partition(a,b), (xis: (xis?(a,b,P)))):
%                                Rie_sum(a,b,P,xis,f) = Rie_sum_alt(a,b,P,xis,f)

%    Riemann_sum?(a:T,b:{x:T|a<x},P:partition(a,b),f:[T->real])(S:real): bool =
%      (EXISTS (xis: (xis?(a,b,P))): LET N = length(P)-1 IN
%            S = Rie_sum(a,b,P,xis,f))


    integral?(a:T,b:{x:T|a<x},f:[T->real],S:real): bool % =
   %               (FORALL (epsi: posreal): (EXISTS (delta: posreal):
   %                  (FORALL (P: partition(a,b)):
   %                      width(a,b,P) < delta IMPLIES
   %                         (FORALL (R: (Riemann_sum?(a,b,P,f))):
   %                              abs(S - R) < epsi))))

   % x_in(aa:T,bb:{x:T|aa<x}): {t: T | aa <= t AND t <= bb}

   % pick_one(a:T,b:{x:T|a<x},P: partition(a,b))(ii: below(length(P)-1)): real =
   %                             x_in(P(ii), P(ii+1))

   % gen_xis(a:T,b:{x:T|a<x},P: partition(a,b)): (xis?(a,b,P)) =
   %                                   (# length := length(P) - 1,
   %                                      seq :=  pick_one(a,b,P)
   %                                    #)

   % N: VAR above(1)

   % len_eq_part  : LEMMA a < b IMPLIES length(eq_partition(a,b,N)) = N
   % eq_part_lem_a: LEMMA a < b IMPLIES seq(eq_partition(a,b,N))(0) = a
   % eq_part_lem_b: LEMMA a < b IMPLIES seq(eq_partition(a,b,N))(N-1) = b

   % width_eq_part: LEMMA a < b IMPLIES
   %                          width(a,b,eq_partition(a,b,N)) = (b-a)/(N-1)

   % delta: VAR posreal
   % N_from_delta: LEMMA a < b IMPLIES
   %                        LET N = 2 + floor((b - a) / delta),
   %                            EP = eq_partition(a, b, N) IN
   %                        width(a, b, EP) < delta

   % Riemann?_Rie: LEMMA a < b IMPLIES FORALL (P: partition(a,b),
   %                                           xis: (xis?(a,b,P))):
   %                             Riemann_sum?(a,b,P,f)(Rie_sum(a,b,P,xis,f))

   % AUTO_REWRITE+ Riemann?_Rie

   % A1, A2: VAR real
   % integral_unique: LEMMA a < b AND integral?(a,b,f,A1) AND
   %                           integral?(a,b,f,A2) IMPLIES A1 = A2

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

    integral(a:T,b:{x:T|a<x}, ff: { f | integrable?(a,b,f)} ):
                     {S: real | integral?(a,b,ff,S)}

   % s: VAR real
   % integral_def: LEMMA a < b IMPLIES
   %                 ( (integrable?(a,b,f) AND integral(a,b,f) = s)
   %                        IFF integral?(a,b,f,s) )

    integral_restrict_eq: AXIOM a < b AND
                                 (FORALL x: a <= x AND x <= b IMPLIES
                                           f(x) = g(x)) AND
                                integrable?(a,b,f)
                           IMPLIES integrable?(a,b,g) AND
                                    integral(a,b,g) = integral(a,b,f)

   Integrable?(a:T,b:T,f:[T->real]): bool % = (a = b) OR
  %                               (a < b AND integrable?(a,b,f)) OR
  %                               (b < a  AND integrable?(b,a,f))

   Integrable_funs(a,b): TYPE = { f | Integrable?(a,b,f)}

   % Integral?(a:T,b:T,f:[T->real],S:real): bool = (a = b AND S = 0) OR
   %                             (a < b AND integral?(a,b,f,S))


   Integral(a:T,b:T,f:Integrable_funs(a,b)): real % =
%                        IF a = b THEN 0
%                        ELSIF a < b THEN integral(a,b,f)
%                        ELSE -integral(b,a,f)
%                       ENDIF

   Integrable?_rew   : AXIOM a < b AND Integrable?(a,b,f) IMPLIES integrable?(a,b,f)
   Integral_rew      : AXIOM a < b AND Integrable?(a,b,f) IMPLIES Integral(a,b,f) = integral(a,b,f)

END integral_def

%  Note: The definition of Riemann_sum:
%
%
%     Rie_sum(a:T,b:{x:T|a<x},P:partition(a,b),
%      xis: (xis?(a,b,P)),f:[T->real]): real =
%      LET N = length(P)-1 IN
%      sigma[below(N)](0,N-1,(LAMBDA (n: below(N)):
%   (P(n+1) - P(n)) * f(xis(n))))
%
%  uses sigma[below] rather than sigma[nat].  This produces some
%  additional tedium, but makes it unnecessary to surround the
%  LAMBDA expression with an IF for type-correctness
%
%   Rie_sum(a:T,b:{x:T|a<x},P:partition(a,b),
%                  xis: (xis?(a,b,P)),f:[T->real]): real =
%                  LET N = length(P)-1 IN
%          sigma[nat](0,N-1,(LAMBDA (n: nat): IF n < N THEN
%                                       (P(n+1) - P(n)) * f(xis(n))
%                                             ELSE 0
%                                             ENDIF))

quality93%

¤ Dauer der Verarbeitung: 0.13 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.