Certification Problem
Input (COPS 963)
We consider the TRS containing the following rules:
|
b(a(a(x))) |
→ |
a(x) |
(1) |
|
a(a(a(x))) |
→ |
b(b(b(x))) |
(2) |
|
b(b(x)) |
→ |
a(b(a(x))) |
(3) |
The underlying signature is as follows:
{b/1, a/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
|
= |
b(a(a(b(a(a(x10)))))) |
|
→
|
b(a(a(a(x10)))) |
|
→
|
b(b(b(b(x10)))) |
|
→
|
b(a(b(a(b(x10))))) |
|
= |
t3
|
| t0
|
= |
b(a(a(b(a(a(x10)))))) |
|
→
|
a(b(a(a(x10)))) |
|
→
|
a(a(x10)) |
|
= |
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.