Certification Problem
Input (COPS 587)
We consider the TRS containing the following rules:
|
+(0,y) |
→ |
y |
(1) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
+(x,+(y,z)) |
→ |
+(+(x,y),z) |
(3) |
The underlying signature is as follows:
{+/2, 0/0, s/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2022)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
+(s(x),+(0,x41)) |
|
→
|
+(s(x),x41) |
|
→
|
s(+(x,x41)) |
|
= |
t2
|
| t0
|
= |
+(s(x),+(0,x41)) |
|
→
|
s(+(x,+(0,x41))) |
|
→
|
s(+(+(x,0),x41)) |
|
= |
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.