Certification Problem
Input (COPS 974)
We consider the TRS containing the following rules:
|
0(1(2(3(4(5(1(x))))))) |
→ |
1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x))))))))))))))))))) |
(1) |
|
0(1(2(3(4(5(1(x))))))) |
→ |
1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x))))))))))))))))))))))))) |
(2) |
The underlying signature is as follows:
{0/1, 1/1, 2/1, 3/1, 4/1, 5/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2023)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
0(1(2(3(4(5(1(x))))))) |
|
→
|
1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x))))))))))))))))))) |
|
= |
t1
|
| t0
|
= |
0(1(2(3(4(5(1(x))))))) |
|
→
|
1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x))))))))))))))))))))))))) |
|
= |
t1
|
The two resulting terms cannot be joined for the following reason:
- When applying the cap-function on both terms (where variables may be treated like constants)
then the resulting terms do not unify.