Certification Problem
Input (COPS 956)
We consider the TRS containing the following rules:
|
r(e(x)) |
→ |
w(r(x)) |
(1) |
|
i(t(x)) |
→ |
e(r(x)) |
(2) |
|
e(w(x)) |
→ |
r(i(x)) |
(3) |
|
t(e(x)) |
→ |
r(e(x)) |
(4) |
|
w(r(x)) |
→ |
i(t(x)) |
(5) |
|
e(r(x)) |
→ |
e(w(x)) |
(6) |
|
r(i(t(e(r(x))))) |
→ |
e(w(r(i(t(e(x)))))) |
(7) |
The underlying signature is as follows:
{r/1, e/1, w/1, i/1, t/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
|
= |
e(r(e(x355))) |
|
→
|
e(w(r(x355))) |
|
→
|
r(i(r(x355))) |
|
= |
t2
|
| t0
|
= |
e(r(e(x355))) |
|
→
|
e(w(e(x355))) |
|
→
|
r(i(e(x355))) |
|
= |
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.