Certification Problem
Input (COPS 32)
We consider the TRS containing the following rules:
|
f(g(x),h(x)) |
→ |
a |
(1) |
|
g(b) |
→ |
d |
(2) |
|
h(c) |
→ |
d |
(3) |
The underlying signature is as follows:
{f/2, g/1, h/1, a/0, b/0, d/0, c/0}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
|
= |
f(g(b),h(b)) |
|
→
|
f(d,h(b)) |
|
= |
t1
|
| t0
|
= |
f(g(b),h(b)) |
|
→
|
a |
|
= |
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.