Certification Problem
Input (COPS 584)
We consider the TRS containing the following rules:
|
br(0,y,z) |
→ |
y |
(1) |
|
br(s(x),y,z) |
→ |
z |
(2) |
|
p(0) |
→ |
0 |
(3) |
|
p(s(x)) |
→ |
x |
(4) |
|
+(x,y) |
→ |
br(x,y,+(p(x),s(y))) |
(5) |
|
+(x,y) |
→ |
br(y,x,+(s(x),p(y))) |
(6) |
The underlying signature is as follows:
{br/3, 0/0, s/1, p/1, +/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
|
= |
+(x,y) |
|
→
|
br(x,y,+(p(x),s(y))) |
|
= |
t1
|
| t0
|
= |
+(x,y) |
|
→
|
br(y,x,+(s(x),p(y))) |
|
= |
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.