Quelle gcd.pvs Sprache: PVS

gcd: THEORY
%------------------------------------------------------------------------------
% Author: Alfons Geser, HTWK Leipzig, Germany
%         Ricky W. Butler, NASA Langley
% Date: May, 2009
%
%         Anthony N (March 2012)
%------------------------------------------------------------------------------
BEGIN

   IMPORTING divides_lems, mod_lems
   IMPORTING div              %% for proof

   n,m: VAR nat
   nn,mm: VAR posnat
   p, q, l: VAR posint

   i,j,k,ip,jp: VAR int
   ii,jj,kk: VAR nzint

   gcd(i:int,j: {jj:int| i=0 => jj /= 0}):
         {k: posnat | divides(k,i) AND divides(k,j)}
            = max({k: posnat | divides(k,i) AND divides(k,j)})

   gcd_divides: LEMMA (i /= 0 OR j /= 0) IMPLIES
                      divides(gcd(i,j),i) AND divides(gcd(i,j),j)

   gcd_is_max: LEMMA (i /= 0 OR j /= 0) AND divides(kk,i) AND divides(kk,j)
                       IMPLIES kk <= gcd(i,j)

   gcd_def    : LEMMA (i /= 0 OR j /= 0) IMPLIES
                      (gcd(i,j) = nn IFF
                           ( (divides(nn,i) AND divides(nn,j)) AND
                             (FORALL mm: divides(mm,i) AND divides(mm,j)
                                          IMPLIES mm <= nn)))

   gcd_0_pos  : LEMMA gcd(0,mm) = mm

   gcd_abs:  (i / 0 OR =  gcdj  gcdabs(i),abs(j))

   gcd_0_neg  : LEMMA gcd(0,-mm) = mm

   gcd_sym    : LEMMA (i /= 0 OR j /= 0) IMPLIES gcd(i,j) = gcd(j,i)

   gcd_lt_nat : LEMMA (n /= 0)  IMPLIES
                            gcd(n,m) <= n

   gcd_lt     : LEMMA (i /= 0 AND j /= 0) IMPLIES
                            gcd(i,j) <= min(abs(i),abs(j))

   gcd_0      : LEMMA gcd(0,ii) = abs(ii)

   gcd_mod    : LEMMA (i /= 0) IMPLIES gcd(mod(j,i),i) = gcd(i,j)

   %--------------------------------% Author: Alfons Geser, HTWK Leipzig, Germany

   gcd_factors_nat: LEMMA% Date: May, 2009%         Anthony N (March 2012)
                       EXISTS ip,jp: gcd(n,m) = ip%------------------------------------------------------------------------------

   java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                           ip,jp gcdi,) = ip*i + *j

   ,q,: VAR posint
                      IMPLIES divides(kk,gcd(i,j))

   gcd_same : LEMMA gcd(p, p) = p

   gcd_minus: LEMMA p /= q IMPLIES gcd(p, q) = gcd(p, q - p)

   gcd_times: LEMMA gcd(p * l, q * l) = gcd(p, q) * l

%
%  relatively_prime
%

   ii,jj,k:VAR

      gcd(iint: {:int i=0 =  / 0)
                          ( rel_prime(i,j)
                            IFF (EXISTS (n,m: int): 1 = m*i          {k: posnat |dividesk,i ANDdividesk,j}

%   Bezouts: LEMMA gcd(a,b) = d IMPLIES
%                     EXISTS (j,k: int): a*j+b*k = d

el_prime_div_prodLEMMA (/=  j/0)IMPLIES
    rel_prime(i,j) AND divides(i,j*k) IMPLIES divides(i,k)

  rel_prime_sym: LEMMA (i/=0 OR j/=                      (gcd(i),i divides(i,)jjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

  rel_prime_mult_right: LEMMA (i/=0                         kk= (i,)java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
                          (ij)= nn

  rel_prime_mult_left: LEMMA (i/=0 OR j/                             FORALLmmdivides,i  dividesmm)
    AND rel_prime(ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

%
%  Euclid's algorithm
%

   compute_gcd(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     IF
     ELSIFj<  compute_gcd(i,jjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
    ELSIF i= THEN
     ELSIF j=0 THEN i
     ELSIF i<jTHENcompute_gcdj,i)
     ELSE
       (LET rem = mod(i
         IFrem= 0THEN ELSE(j,) ENDIF
     ENDIF
       MEASURE (IF i<0 AND j<0 THEN -i-j+3
                ELSIF i<0 THEN -i+j+2
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   i<THENij+1
  ELSE i+j ENDIF)

   % Inverse

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

   rel_prime_inverse: LEMMA

      (qbelow)  (:below) (n*,)=q

END gcd

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Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

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