Certification Problem
Input (COPS 569)
We consider the TRS containing the following rules:
|
s(p(x)) |
→ |
x |
(1) |
|
p(s(x)) |
→ |
x |
(2) |
|
+(x,0) |
→ |
x |
(3) |
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(4) |
|
+(x,p(y)) |
→ |
p(+(x,y)) |
(5) |
|
-(x,0) |
→ |
x |
(6) |
|
-(x,s(y)) |
→ |
p(-(x,y)) |
(7) |
|
-(x,p(y)) |
→ |
s(-(x,y)) |
(8) |
|
*(x,0) |
→ |
0 |
(9) |
|
*(x,s(y)) |
→ |
+(*(x,y),x) |
(10) |
|
*(x,p(y)) |
→ |
-(*(x,y),x) |
(11) |
The underlying signature is as follows:
{s/1, p/1, +/2, 0/0, -/2, */2}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
|
= |
*(x,s(p(x379))) |
|
→
|
*(x,x379) |
|
= |
t1
|
| t0
|
= |
*(x,s(p(x379))) |
|
→
|
+(*(x,p(x379)),x) |
|
= |
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.