Certification Problem
Input (COPS 13)
We consider the TRS containing the following rules:
|
f(f(x,y),z) |
→ |
f(x,f(y,z)) |
(1) |
|
f(x,1) |
→ |
x |
(2) |
The underlying signature is as follows:
{f/2, 1/0}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(f(f(x31,1),y),z) |
|
→
|
f(f(x31,y),z) |
|
→
|
f(x31,f(y,z)) |
|
= |
t2
|
| t0
|
= |
f(f(f(x31,1),y),z) |
|
→
|
f(f(x31,1),f(y,z)) |
|
→
|
f(x31,f(1,f(y,z))) |
|
= |
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.