/* 
PROOF OF CORRECTNESS:
  DIV(X,Y): compute X / Y (integer) and  X % Y (remainder or modulo)

Trace: div(5,2) = 2 with remainder 1

  x   y   z   z*Y+x
-------------------
  5   2   0     5
  3   2   1     5
  1   2   2     5

As the loop iterates, z*Y+x stays constant at 5 (X). This is an unusual situation
and results in a differnt looking INVARIANT: X=z*Y+x

This diagram shows how the algorithm works:

   +--------- x=X=5
  5|
   +          subtract y=2, z=1= # of times y has been subtracted
  4|
   +---------
  3|
   +          subtract y=2, z=2= # of times y has been subtracted
  2|
   +---------
  1|          cannot subtract y=2 anymore, x=1 is now the remainder
   +
  0+--------- 

1. start x at X=5, keep subtracting y=2 from x
2. stop when it is not possible to subtract anymore y, x is the remainder

Back to the INVARIANT: X=z*Y+x which says that z represents how many times it has substracted
  (or Y) from x. In the beginning, z*Y is zero, and all the weight is on x. As the loop 
iterates, the weight is transferred from x onto z*Y. But they always stay in balance:
X is always equal to the amounts of Y's that have been substracted plus the remainder x.
The final result has subtracted as much as possible and x must be < y.
*/
#include <stdio.h>

int div(int X, int Y) {
int x, y, z;

  // PRE: X > 0 and Y > 0
  
  x = X;
  // x = X and x > 0
  
  y = Y;
  // x = X and x > 0 and y = Y and y > 0
  
  z = 0;
  // x = X and x > 0 and y = Y and y > 0 and z = 0
  
  // INVARIANT: X = z * Y + x and x >= 0 and y = Y
  while (x >= y) {
    
    // X = z * Y + x and x >= 0 and y = Y and x >= y
    
    z = z + 1;
    
    // X = (z-1) * Y + x and x >= 0 and y = Y and x >= y
    // X = z * Y - Y + x and x >= 0 and y = Y and x >= y
    // X = z * Y + x - Y and x >= 0 and y = Y and x >= y
    // X = z * Y + x - y and x >= 0 and y = Y and x >= y
    
    x = x - y;
    
    // X = z * Y + x and x >= 0 and y = Y
  } // end while
  
  // INVARIANT: X = z * Y + x and x >= 0 and y = Y and EXIT: x < y
  // X = z * Y + x and 0 <= x < Y     // x must be 0..Y-1 to be a modulo
  // POST: z = Div(X,Y) and x = Mod(X,Y)

  return z;
}

int main() {
  printf("div(5,2)=%d\n",div(5,2));
}