Certification Problem
Input (COPS 10)
We consider the TRS containing the following rules:
|
f(g(f(x))) |
→ |
x |
(1) |
|
f(g(x)) |
→ |
g(f(x)) |
(2) |
The underlying signature is as follows:
{f/1, g/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2021)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
f(g(f(f(g(f(x5)))))) |
|
→
|
f(g(f(x5))) |
|
→
|
g(f(f(x5))) |
|
= |
t2
|
| t0
|
= |
f(g(f(f(g(f(x5)))))) |
|
→
|
f(g(f(x5))) |
|
→
|
x5 |
|
= |
t2
|
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.