Certification Problem
Input (COPS 590)
We consider the TRS containing the following rules:
|
+(0,0) |
→ |
0 |
(1) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
+(x,s(y)) |
→ |
s(+(y,x)) |
(3) |
|
s(s(x)) |
→ |
x |
(4) |
The underlying signature is as follows:
{+/2, 0/0, s/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
|
= |
+(s(x111),s(y)) |
|
→
|
s(+(x111,s(y))) |
|
→
|
s(s(+(y,x111))) |
|
→
|
+(y,x111) |
|
= |
t3
|
| t0
|
= |
+(s(x111),s(y)) |
|
→
|
s(+(y,s(x111))) |
|
→
|
s(s(+(x111,y))) |
|
→
|
+(x111,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.