Certification Problem
Input (COPS 957)
We consider the TRS containing the following rules:
|
a(b(x)) |
→ |
c(d(x)) |
(1) |
|
d(d(x)) |
→ |
b(e(x)) |
(2) |
|
b(x) |
→ |
d(c(x)) |
(3) |
|
d(x) |
→ |
x |
(4) |
|
e(c(x)) |
→ |
d(a(x)) |
(5) |
|
a(x) |
→ |
e(d(x)) |
(6) |
The underlying signature is as follows:
{a/1, b/1, c/1, d/1, e/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2022)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
d(d(x140)) |
|
→
|
b(e(x140)) |
|
→
|
d(c(e(x140))) |
|
→
|
c(e(x140)) |
|
= |
t3
|
| t0
|
= |
d(d(x140)) |
|
→
|
d(x140) |
|
→
|
x140 |
|
= |
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.