Certification Problem
Input (COPS 635)
We consider the TRS containing the following rules:
|
plus(0,y) |
→ |
y |
(1) |
|
plus(s(x),y) |
→ |
s(plus(x,y)) |
(2) |
|
qplus(0,y) |
→ |
y |
(3) |
|
qplus(s(x),y) |
→ |
qplus(x,s(y)) |
(4) |
|
plus(x,y) |
→ |
qplus(x,y) |
(5) |
The underlying signature is as follows:
{plus/2, 0/0, s/1, qplus/2}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
|
= |
plus(s(plus(0,x67)),y) |
|
→
|
plus(s(x67),y) |
|
→
|
qplus(s(x67),y) |
|
→
|
qplus(x67,s(y)) |
|
= |
t3
|
| t0
|
= |
plus(s(plus(0,x67)),y) |
|
→
|
s(plus(plus(0,x67),y)) |
|
→
|
s(qplus(plus(0,x67),y)) |
|
→
|
s(qplus(qplus(0,x67),y)) |
|
= |
t3
|
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.