Certification Problem
Input (COPS 943)
We consider the TRS containing the following rules:
|
a(a(x)) |
→ |
b(c(x)) |
(1) |
|
b(b(x)) |
→ |
c(d(x)) |
(2) |
|
c(c(x)) |
→ |
d(d(d(x))) |
(3) |
|
d(d(d(x))) |
→ |
a(c(x)) |
(4) |
The underlying signature is as follows:
{a/1, b/1, c/1, d/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
|
= |
a(a(a(x37))) |
|
→
|
a(b(c(x37))) |
|
= |
t1
|
| t0
|
= |
a(a(a(x37))) |
|
→
|
b(c(a(x37))) |
|
= |
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.