Proving Programs Examples

Prove each of the following code segments are correct by backing up the postcondition to get an appropriate precondition 

1. read(x);
   x := x * x;
   x := x * x;
   x := x * x;
   write(x);
   {  output = input ^ 8   }

2. read(x);
   if (x < 0) then
       x := 0 - x
   write(x);
   { output >= 0 and (output = input or output = 0 - input}

3. read(x);
   res := 1;
   while (x > 0) do BEGIN
       res := res * 2;
       x := x - 1;
   END;
   write(res);
   { output = 2^input }

4. read(x);
   read(y);
   res := 0;
   while (y > 0) do BEGIN
        res := res + x;
        y := y - 1;
   END;
   write(res);
   {  output = input1 * input2  }

5. This one is a little tricky ...
     You can assume that all variables are integers.
   read(x);
   res := 0;
   y := 0
   while (x > y) do BEGIN
        y := y + 1;
        res := res + y;
   END;
   write(res);
   {  output = input * (input + 1) / 2  }