Overture Tool

/**
 * A specification of the Luhn check digit algorithm.
 */
types
  Digit = nat      -- A decimal digit, 0-9
  inv d == d < 10;
  
functions
  luhn: seq1 of Digit -> Digit  -- Non empty list input
  luhn(data) ==
    total(data) * 9 mod 10;
  
  -- Convenience function for "12345"
  luhns: seq1 of char -> Digit
  luhns(number) ==
    luhn(strToSeq(number));
  
  -- Convenience function for numbers
  luhnn: nat -> Digit
  luhnn(number) ==
    luhn(natToSeq(number));
    
  total: seq of Digit -> nat
  total(data) ==
    if data = []
    then
      0
    else
      let multipler = (len data) mod 2 + 1,
        product = hd data * multipler
      in
        total(tl data) +  -- recurse
          if product < 10
          then product
          else (product mod 10) + 1
  measure len data;
      
  strToSeq: seq1 of char -> seq1 of Digit
  strToSeq(s) ==
    [ cases i :
      '0'  -> 0, '1' -> 1, '2' -> 2, '3' -> 3, '4' -> 4,
      '5'  -> 5, '6' -> 6, '7' -> 7, '8' -> 8, '9' -> 9
      end | i in seq s];

  natToSeq: nat -> seq of Digit
  natToSeq(n) ==
    if n < 10
    then [n]
    else natToSeq(n div 10) ^ [n rem 10]
  measure n;

traces
  /**
   * Generate all the possible 1, 2 and 3 digit sequences and check that the luhn
   * calculation completes without breaking any constraints.
   */
  First1000:
    let a,b,c,d in set {0,...,9} in
    (
      luhn([a]);
      luhn([a,b]);
      luhn([a,b,c]);
      luhn([a,b,c,d])
    );
    
  /**
   * The Luhn algorithm will detect any single-digit error, as well as almost all
   * transpositions of adjacent digits. It will not, however, detect transposition
   * of the two-digit sequence 09 to 90 (or vice versa).
   *
   * See http://en.wikipedia.org/wiki/Luhn_algorithm
   */
  AllOneDigitErrors:
    let input = [7,9,9,2,7,3,9,8,7,1] in
    let pos in set inds input in
    let replacement in set {0,...,9} \ {input(pos)} in
    let corrupt = input(1,...,pos-1) ^ [replacement] ^ input(pos+1,...,len input) in
      checkFail(corrupt, 3);

  AllAdjacentTranspositions:
    let input = [7,9,9,2,7,3,9,8,7,1] in
    let pos in set inds tl input be st  -- ie. one less that the length
      input(pos+1) <> input(pos)
      and {input(pos+1), input(pos)} <> {0,9} in
    let replacement = [input(pos+1), input(pos)] in
    let corrupt = input(1,...,pos-1) ^ replacement ^ input(pos+2,...,len input) in
      checkFail(corrupt, 3);

--  AllTwoDigitErrors:
--    let input = [7,9,9,2,7,3,9,8,7,1] in
--    let p1, p2 in set inds input in
--    let r1 in set {0,...,9} \ {input(p1)} in
--    let r2 in set {0,...,9} \ {input(p2)} in
--    let first  = input(1,...,p1-1) ^ [r1] ^ input(p1+1,...,len input) in
--    let second = first(1,...,p2-1) ^ [r2] ^ first(p2+1,...,len first) in
--      checkFail(second, 3);
  
  /**
   * It will detect 7 of the 10 possible twin errors (it will not detect
   * 22 <> 55, 33 <> 66 or 44 <> 77).
   *
   * See http://en.wikipedia.org/wiki/Luhn_algorithm
   */
  AllTwinErrors:
    let input = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,0,0] in
    let pos in set inds tl input be st input(pos) = input(pos+1) in
    let rep in set {0,...,9} \ {input(pos)} in
    let corrupt = input(1,...,pos-1) ^ [rep, rep] ^ input(pos+2,...,len input) in
      checkFail(corrupt, 0);
  
  /**
   * Because the algorithm operates on the digits in a right-to-left manner and zero
   * digits affect the result only if they cause shift in position, zero-padding the
   * beginning of a string of numbers does not affect the calculation.
   *
   * See http://en.wikipedia.org/wiki/Luhn_algorithm
   */
  ZeroPadding:
    let input = [7,9,9,2,7,3,9,8,7,1] in
    let number in set {1, ..., 10} in
    let padding = [p-p | p in set {1, ..., number}] in 
      checkOK(padding ^ input, 3);
  
operations
  /**
   * These operations support the traces above
   */
  checkFail: seq1 of Digit * Digit ==> bool
  checkFail(data, expected) ==
    return luhn(data) <> expected  -- Expect failure!
  post RESULT = true;
  
  checkOK: seq1 of Digit * Digit ==> bool
  checkOK(data, expected) ==
    return luhn(data) = expected  -- Expect success!
  post RESULT = true;