Certification Problem
Input (COPS 762)
We consider the TRS containing the following rules:
| a |
→ |
b |
(1) |
|
f(a,b) |
→ |
f(a,a) |
(2) |
|
f(b,a) |
→ |
f(a,a) |
(3) |
|
f(a,a) |
→ |
c |
(4) |
|
g(x) |
→ |
f(x,x) |
(5) |
The underlying signature is as follows:
{a/0, b/0, f/2, c/0, g/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
|
= |
g(a) |
|
→
|
g(b) |
|
→
|
f(b,b) |
|
= |
t2
|
| t0
|
= |
g(a) |
|
→
|
f(a,a) |
|
→
|
c |
|
= |
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.