Certification Problem
Input (COPS 630)
We consider the TRS containing the following rules:
|
min(0,y) |
→ |
0 |
(1) |
|
min(x,0) |
→ |
0 |
(2) |
|
min(s(x),s(y)) |
→ |
min(x,y) |
(3) |
|
min(x,x) |
→ |
x |
(4) |
The underlying signature is as follows:
{min/2, 0/0, s/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
|
= |
min(s(x97),s(x97)) |
|
→
|
min(x97,x97) |
|
→
|
x97 |
|
= |
t2
|
| t0
|
= |
min(s(x97),s(x97)) |
|
→
|
s(x97) |
|
= |
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.