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Datei: real_continuity.pvs Sprache: PVS

Original von: PVS^©

%------------------------------------------------------------------------------
% Continuous functions [T->real]
%
%     Author: David Lester, Manchester University
%
%     Version 1.0            15/09/08  Initial Version
%------------------------------------------------------------------------------

real_continuity[T:TYPE,(IMPORTING metric_def[T]) d:metric]: THEORY

BEGIN

  IMPORTING
    metric_continuity[T,d,real,(lambda (x,y:real): abs(x-y))],
    topology@constant_continuity[T,metric_induced_topology[T,d],
                                 real,metric_induced_topology[real,
                                            (lambda (x,y:real) :abs(x-y))]],
    reals@real_fun_ops_aux[T],
    power@real_expt,
    power@log,
    power@real_fun_power[T]

  g,g1,g2: VAR metric_continuous
  n: VAR nat
  u: VAR sequence[metric_continuous]
  c: VAR real
  a: VAR posreal
  x: VAR T
  h: VAR {g | FORALL x: g(x) >= 0}

%;
%  ^(g,a): MACRO [T->real] = real_fun_power.^(g,a);

  sum_continuous:       JUDGEMENT +(g1,g2)     HAS_TYPE metric_continuous
  opp_continuous:       JUDGEMENT -(g)         HAS_TYPE metric_continuous
  diff_continuous:      JUDGEMENT -(g1,g2)     HAS_TYPE metric_continuous
  scal_continuous:      JUDGEMENT *(c,g)       HAS_TYPE metric_continuous
  abs_continuous:       JUDGEMENT abs(g)       HAS_TYPE metric_continuous
  expt_real_continuous: JUDGEMENT ^(h,a)       HAS_TYPE metric_continuous
  expt_nat_continuous:  JUDGEMENT expt(g,n)    HAS_TYPE metric_continuous
  sq_continuous:        JUDGEMENT sq(g)        HAS_TYPE metric_continuous
  min_continuous:       JUDGEMENT min(g1,g2)   HAS_TYPE metric_continuous
  max_continuous:       JUDGEMENT max(g1,g2)   HAS_TYPE metric_continuous
  minimum_continuous:   JUDGEMENT minimum(u,n) HAS_TYPE metric_continuous
  maximum_continuous:   JUDGEMENT maximum(u,n) HAS_TYPE metric_continuous
  plus_continuous:      JUDGEMENT plus(g)      HAS_TYPE metric_continuous
  minus_continuous:     JUDGEMENT minus(g)     HAS_TYPE metric_continuous
  prod_continuous:      JUDGEMENT *(g1,g2)     HAS_TYPE metric_continuous

END real_continuity

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