Certification Problem
Input (COPS 261)
We consider the TRS containing the following rules:
|
-(x,x) |
→ |
0 |
(1) |
|
-(s(x),s(y)) |
→ |
-(x,y) |
(2) |
|
+(x,y) |
→ |
+(y,x) |
(3) |
|
+(0,x) |
→ |
x |
(4) |
|
+(x,0) |
→ |
x |
(5) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(6) |
|
+(x,s(y)) |
→ |
s(+(y,x)) |
(7) |
|
+(p(x),y) |
→ |
p(+(x,y)) |
(8) |
|
+(x,p(y)) |
→ |
p(+(y,x)) |
(9) |
|
s(p(x)) |
→ |
x |
(10) |
|
p(s(x)) |
→ |
x |
(11) |
The underlying signature is as follows:
{-/2, 0/0, s/1, +/2, p/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
|
= |
-(s(p(x414)),s(y)) |
|
→
|
-(x414,s(y)) |
|
= |
t1
|
| t0
|
= |
-(s(p(x414)),s(y)) |
|
→
|
-(p(x414),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.