Quelle derivatives_def.pvs Sprache: PVS

derivatives_def  [ T : TYPE FROM real ] : THEORY
%-----------------------------------------------------------------------------------------
%
%    The derivatives library has been generalized to handle a larger class of domains.
%    Previously the domain had to be a connected domain defined by the following
%    predicate:
%
%       connected_domain : ASSUMPTION
%           FORALL (x, y : T), (z : real) :
%               x <= z AND z <= y IMPLIES T_pred(z)
%
%    The following more general domain definition is now used
%
%        deriv_domain    : ASSUMPTION FORALL (e: posreal, x:T):
%                                        EXISTS (y: {u: nzreal | T_pred(u + x)}): abs(y) < e
%
%    This definition allows unions of closed and open intervals.
%    See deriv_domains for more information.
%
%-----------------------------------------------------------------------------------------
BEGIN

  ASSUMING
    IMPORTING deriv_domain_def

    deriv_domain     : ASSUMPTION deriv_domain?[T]

    not_one_element  : ASSUMPTION not_one_element?[T]

  ENDASSUMING

  IMPORTING lim_of_functions, continuous_functions

  f, f1, f2, fp : VAR [T -> real]
  g : VAR [T -> nzreal]
  x : VAR T
  u : VAR nzreal
  b : VAR real
  l, l1, l2 : VAR real

  %-------------------
  %  Newton Quotient
  %-------------------

  A(x) : setof[nzreal] = { u: nzreal | T_pred(x + u) }

  NQ(f, x)(h : (A(x))) : real = (f(x + h) - f(x)) / h

  deriv_TCC : LEMMA FORALL x : adh[(A(x))](fullset[real])(0)
  adh_A_lem : LEMMA adh[(A(x))](fullset[real])(0)   %% Butler

%   play: LEMMA (FORALL (x:T): EXISTS (e: posreal):
%      FORALL (y: real): abs(x-y) < e IMPLIES T_pred(y))
%               IMPLIES
%                  adh[(A(x))](fullset[real])(0)

  %----------------------------
  %  Differentiable functions
  %----------------------------

  derivable?(f, x) : bool = convergent?(NQ(f, x), 0)

%% +++  derivable?(f) : bool = FORALL x : derivable?(f, x)

  %--------------------------------------
  %  Derivable functions are continuous
  %--------------------------------------

  continuous_lim  : LEMMA convergence(LAMBDA (h : (A(x))) : f(x + h), 0, f(x))
                            IFF continuous?(f, x)

  deriv_continuous      : LEMMA convergence(NQ(f, x), 0, l)
                                  IMPLIES continuous?(f, x)

  derivable_continuous  : LEMMA derivable?(f, x) IMPLIES continuous?(f, x)

%% +++  derivable_cont_fun : LEMMA derivable?(f) IMPLIES continuous?(f)

  %---------------------
  %  Properties of NQ
  %---------------------

  sum_NQ      : LEMMA NQ(f1 + f2, x) = NQ(f1, x) + NQ(f2, x)

  neg_NQ      : LEMMA NQ(- f, x) = - NQ(f, x)

  diff_NQ     : LEMMA NQ(f1 - f2, x) = NQ(f1, x) - NQ(f2, x)

  scal_NQ     : LEMMA NQ(b * f, x) = b * NQ(f, x)

  const_NQ    : LEMMA NQ(const_fun(b), x) = const_fun(0)

  identity_NQ : LEMMA NQ(I[T], x) = const_fun(1)

  prod_NQ     : LEMMA FORALL (h : (A(x))): NQ(f1 * f2, x)(h)
                            = NQ(f1, x)(h) * f2(x) + NQ(f2, x)(h) * f1(x + h)

  cnv_seq_prod_NQ: LEMMA convergence(NQ(f1, x), 0, l1)
                          AND convergence(NQ(f2, x), 0, l2)
              IMPLIES convergence(NQ(f1 * f2, x), 0, f2(x) * l1 + f1(x) * l2)

  inv_NQ         : LEMMA FORALL (h : (A(x))) : NQ(1/g, x)(h)
                                   = - NQ(g, x)(h) / (g(x) * g(x + h))

  cnv_seq_inv_NQ : LEMMA convergence(NQ(g, x), 0, l1)
                       IMPLIES convergence(NQ(1/g, x), 0, - l1 / (g(x) * g(x)))

  %---------------------------------------
  %  Operations preserving derivability
  %---------------------------------------

  sum_derivable     : LEMMA derivable?(f1, x) AND derivable?(f2, x)
                               IMPLIES derivable?(f1 + f2, x)

  neg_derivable     : LEMMA derivable?(f, x) IMPLIES derivable?(- f, x)

  diff_derivable    : LEMMA derivable?(f1, x) AND derivable?(f2, x)
                               IMPLIES derivable?(f1 - f2, x)

  prod_derivable    : LEMMA derivable?(f1, x) AND derivable?(f2, x)
                               IMPLIES derivable?(f1 * f2, x)

  scal_derivable    : LEMMA derivable?(f, x) IMPLIES derivable?(b * f, x)

  const_derivable   : LEMMA derivable?(const_fun(b), x)

  inv_derivable     : LEMMA derivable?(g, x) IMPLIES derivable?(1/g, x)

  div_derivable     : LEMMA derivable?(f, x) AND derivable?(g, x)
                               IMPLIES derivable?(f / g, x)

  identity_derivable: LEMMA derivable?(I, x)

  %--------------
  %  Derivative
  %--------------

  deriv(f, (x0 : { x | derivable?(f, x) })) : real = lim(NQ(f, x0), 0)

  deriv_def   : LEMMA convergence(NQ(f, x), 0, l) IMPLIES
                   derivable?(f,x) AND deriv(f,x) = l

  deriv_sum   : LEMMA derivable?(f1, x) AND derivable?(f2, x) IMPLIES
                      deriv(f1 + f2, x) = deriv(f1, x) + deriv(f2, x)

  deriv_neg   : LEMMA derivable?(f, x) IMPLIES
                      deriv(- f, x) = - deriv(f, x)

  deriv_diff  : LEMMA derivable?(f1, x) AND derivable?(f2, x) IMPLIES
                         deriv(f1 - f2, x) = deriv(f1, x) - deriv(f2, x)

  deriv_prod  : LEMMA derivable?(f1, x) AND derivable?(f2, x) IMPLIES
                         deriv(f1 * f2, x) = deriv(f1, x) * f2(x)
                                             + deriv(f2, x) * f1(x)

  deriv_const : LEMMA deriv(const_fun(b), x) = 0

  deriv_scal  : LEMMA derivable?(f, x) IMPLIES deriv(b * f, x) = b * deriv(f, x)

  deriv_inv   : LEMMA derivable?(g, x) IMPLIES
                         deriv(1/g, x) = - deriv(g, x) / (g(x) * g(x))

  deriv_div   : LEMMA derivable?(f, x) AND derivable?(g, x) IMPLIES
                         deriv(f / g, x) =
                               (deriv(f,x)*g(x) - deriv(g,x)*f(x))/(g(x)*g(x))

  deriv_identity : LEMMA deriv(I[T], x) = 1

END derivatives_def

quality92%

¤ 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.0.13Bemerkung: (vorverarbeitet) ¤

*Bot Zugriff

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.