Certification Problem
Input (COPS 961)
We consider the TRS containing the following rules:
|
h(1(1(x))) |
→ |
1(h(x)) |
(1) |
|
1(1(h(b(x)))) |
→ |
1(1(s(b(x)))) |
(2) |
|
1(s(x)) |
→ |
s(1(x)) |
(3) |
|
b(s(x)) |
→ |
b(h(x)) |
(4) |
|
h(1(b(x))) |
→ |
t(1(1(b(x)))) |
(5) |
|
1(t(x)) |
→ |
t(1(1(1(x)))) |
(6) |
|
b(t(x)) |
→ |
b(h(x)) |
(7) |
The underlying signature is as follows:
{h/1, 1/1, b/1, s/1, t/1}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
|
= |
h(1(1(h(b(x288))))) |
|
→
|
h(1(1(s(b(x288))))) |
|
→
|
h(1(s(1(b(x288))))) |
|
→
|
h(s(1(1(b(x288))))) |
|
= |
t3
|
| t0
|
= |
h(1(1(h(b(x288))))) |
|
→
|
1(h(h(b(x288)))) |
|
= |
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.