Certification Problem
Input (COPS 567)
We consider the TRS containing the following rules:
|
f(g(f(x))) |
→ |
g(f(g(x))) |
(1) |
|
f(c) |
→ |
c |
(2) |
|
g(c) |
→ |
c |
(3) |
The underlying signature is as follows:
{f/1, g/1, c/0}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2020)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
f(g(f(f(g(f(x2)))))) |
|
→
|
f(g(f(g(f(g(x2)))))) |
|
→
|
g(f(g(g(f(g(x2)))))) |
|
= |
t2
|
| t0
|
= |
f(g(f(f(g(f(x2)))))) |
|
→
|
g(f(g(f(g(f(x2)))))) |
|
→
|
g(g(f(g(g(f(x2)))))) |
|
= |
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.