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Datei: comparison1.pvs Sprache: PVS

Original von: PVS^©

comparison1
  [b,p:above(1),
   E_max:integer,
   E_min:{i:integer|E_max>i}]: THEORY

  BEGIN

  IMPORTING IEEE_854_values[b,p,E_max,E_min]

  num,num1,num2: var {fp:fp_num| finite?(fp) or infinite?(fp)}

% extend value function so that infinities can be compared to finite
% numbers
  n_value(num): real =
    IF infinite?(num) THEN (-1) ^ i_sign(num) * b ^ (E_max + 1)
    ELSE value(num)
    ENDIF


% From section 5.7 of IEEE-854:
%   ``Four mutually exclusive relations are possible: "less than,"
%  "equal," "greater than," and "unordered."  The last case arises only
%  when at least one operand is a NaN.'' ...

%  ``The result of a comparison shall be delivered in one of two ways at
%  the implementor's option: either as a condition code identifying one
%  of the four relations listed above, or as a true/false response to a
%  predicate that names the specific comparison desired.''

% fp_compare, below, defines the first option:

  comparison_code: type = {gt, lt, eq, un}

  fp_compare((fp1, fp2: fp_num)): comparison_code =
    IF NaN?(fp1) OR NaN?(fp2) THEN un
    ELSIF n_value(fp1) > n_value(fp2) THEN gt
    ELSIF n_value(fp1) < n_value(fp2) THEN lt
    ELSE eq
    ENDIF

  fp,fp1,fp2: var fp_num

  posinf_ge: LEMMA
        gt?(fp_compare(infinite(pos), num))
          OR eq?(fp_compare(infinite(pos), num))

  NaN_unordered: LEMMA
      NaN?(fp1) OR NaN?(fp2) => un?(fp_compare(fp1, fp2))

% option 2, minimal set of predicates is {eq, ne, gt, ge, lt, le} also
% should include un.  These predicates are defined below the level
% where invalid will be signalled.

% The follwing predicates are from table 3, page 12 of IEEE-854.  They
% do not include the signalling of the invalid exception.  That test
% may be applied prior to invoking these predicates.

% shall include
  eq?(fp1,fp2) :bool = eq?(fp_compare(fp1,fp2))
  ne?(fp1,fp2) :bool = not eq?(fp_compare(fp1,fp2))
  gt?(fp1,fp2) :bool = gt?(fp_compare(fp1,fp2))
  ge?(fp1,fp2) :bool = gt?(fp_compare(fp1,fp2)) or eq?(fp_compare(fp1,fp2))
  lt?(fp1,fp2) :bool = lt?(fp_compare(fp1,fp2))
  le?(fp1,fp2) :bool = lt?(fp_compare(fp1,fp2)) or eq?(fp_compare(fp1,fp2))

% should include
  un?(fp1,fp2) :bool = un?(fp_compare(fp1,fp2))

% other
  lg?(fp1,fp2) :bool = gt?(fp_compare(fp1,fp2)) or lt?(fp_compare(fp1,fp2))
  leg?(fp1,fp2):bool = not un?(fp_compare(fp1,fp2))
  ug?(fp1,fp2) :bool = gt?(fp_compare(fp1,fp2)) or un?(fp_compare(fp1,fp2))
  uge?(fp1,fp2):bool = not lt?(fp_compare(fp1,fp2))
  ul?(fp1,fp2) :bool = lt?(fp_compare(fp1,fp2)) or un?(fp_compare(fp1,fp2))
  ule?(fp1,fp2):bool = not gt?(fp_compare(fp1,fp2))
  ue?(fp1,fp2) :bool = eq?(fp_compare(fp1,fp2)) or un?(fp_compare(fp1,fp2))

  eq_def: lemma  eq?(num1,num2) <=> n_value(num1) = n_value(num2)

  ne_def: lemma  ne?(num1,num2) <=> n_value(num1) /= n_value(num2)

  gt_def: lemma  gt?(num1,num2) <=> n_value(num1) > n_value(num2)

  ge_def: lemma  ge?(num1,num2) <=> n_value(num1) >= n_value(num2)

  lt_def: lemma  lt?(num1,num2) <=> n_value(num1) < n_value(num2)

  le_def: lemma  le?(num1,num2) <=> n_value(num1) <= n_value(num2)

  un_def: lemma  un?(fp1,fp2)   <=> (NaN?(fp1) or NaN?(fp2))

  END comparison1

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