This can be proved as follows. 
cannot be chosen at the start, since than the LHS and RHS would differ in the first symbol ( 
in LHS and 
in RHS). So, we must start with 
. The next pair must be so that the 3 rd symbol in the RHS becomes identical to that of the LHS, which is a 
. After this step, LHS and RHS are not matching. If is selected next, then would be mismatched in the 7 th symbol ( 
in LHS and in RHS). If is selected, instead, there will not be any choice to match the both side in the next step.
Example3 : The list 1,3,2,3 is a solution to the following PCP instance.
i |
xi |
yi |
1 |
1 |
101 |
2 |
10 |
00 |
3 |
011 |
11 |
The following properties can easily be proved.
Proposition The Post Correspondence System