Certification Problem
Input (COPS 5)
We consider the TRS containing the following rules:
|
f(g(x,a,b)) |
→ |
x |
(1) |
|
p(a) |
→ |
c |
(2) |
|
g(f(h(c,d)),x,y) |
→ |
h(p(x),q(x)) |
(3) |
|
q(b) |
→ |
d |
(4) |
The underlying signature is as follows:
{f/1, g/3, a/0, b/0, p/1, c/0, h/2, d/0, q/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
|
= |
f(g(f(h(c,d)),a,b)) |
|
→
|
f(h(p(a),q(a))) |
|
= |
t1
|
| t0
|
= |
f(g(f(h(c,d)),a,b)) |
|
→
|
f(h(c,d)) |
|
= |
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.