Certification Problem
Input (COPS 694)
We consider the TRS containing the following rules:
| c |
→ |
b |
(1) |
|
h(b,a) |
→ |
b |
(2) |
| a |
→ |
f(a) |
(3) |
|
h(a,h(c,c)) |
→ |
f(h(b,f(c))) |
(4) |
|
f(h(b,f(c))) |
→ |
f(a) |
(5) |
The underlying signature is as follows:
{c/0, b/0, h/2, a/0, f/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
|
= |
h(a,h(c,c)) |
|
→
|
h(a,h(b,c)) |
|
→
|
h(f(a),h(b,c)) |
|
→
|
h(f(a),h(b,b)) |
|
= |
t3
|
| t0
|
= |
h(a,h(c,c)) |
|
→
|
f(h(b,f(c))) |
|
→
|
f(h(b,f(b))) |
|
= |
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.