Quelle max_array.pvs Sprache: PVS

max_array[N: posnat, T: TYPE, <= : (total_order?[T]) ]: THEORY
%------------------------------------------------------------------------
%
%  max (basic definitions and properties)
%  -------------------------------------------
%
%      Author: Ricky W. Butler
%
%  This theory defines the maximum function over an array of values
%  from an ordered type. This is done in an implementation-independent
%  manner.  However, a recursive executable form is included to
%  demonstrate that the definition is sound.
%
%  The following functions are defined:
%
%      Imax(A,ii,jj)     = general definition of maximum of array A
%                          over the internal range from ii to jj.
%                          Returns the index into the array.
%
%      imax(A)           = general definition of maximum of array A.
%                          Returns the index into the array.
%
%      max(A)            = general definition of maximum of array A.
%                          Returns the maximal element.
%
%  Notes:
%     1. The properties of imax and max are readily obtained via
%        TYPEPRED "max" and TYPEPRED "imax" or through use of
%        max_lem and imax_lem.
%     2. Since there may be replicates in A, the return type of imax
%        may not be a singleton set.
%------------------------------------------------------------------------

  EXPORTING ALL WITH below_arrays[N,T]

BEGIN

  A,A1,A2: VAR ARRAY[below(N) -> T]
  x: VAR T
  ii,jj,j: VAR below(N)

  IMPORTING below_arrays[N,T], max_array_def[N,T,<=]

  imax(A): {i: below(N)| (FORALL ii: A(ii) <= A(i))}

  max(A): { t: T | (FORALL ii: A(ii) <= t) AND
                    (EXISTS jj: A(jj) = t)}

% LEMMAs that follow immediately from TYPEPRED "imax" and TYPEPRED "max"

  max_lem   : LEMMA A(ii) <= max(A)

  max_in?   : LEMMA in?(max(A),A)

  imax_lem  : LEMMA A(imax(A)) = max(A)

  max_def   : LEMMA (FORALL ii: A(ii) <= max(A)) AND in?(max(A),A)

  max_it_is : LEMMA (FORALL ii: A(ii) <= x) AND in?(x,A) IMPLIES
                        max(A) = x

  imax_1    : LEMMA N = 1 IMPLIES imax(A) = 0

  max_2     : LEMMA N = 2 IMPLIES max(A) = IF A(0) <= A(1) THEN A(1)
                                           ELSE A(0) ENDIF

END max_array

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