Certification Problem
Input (COPS 613)
We consider the TRS containing the following rules:
|
+(0,y) |
→ |
y |
(1) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
+(p(x),y) |
→ |
p(+(y,x)) |
(3) |
|
p(s(x)) |
→ |
s(p(x)) |
(4) |
|
s(p(x)) |
→ |
x |
(5) |
The underlying signature is as follows:
{+/2, 0/0, s/1, p/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2020)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
+(p(+(s(x64),x65)),y) |
|
→
|
+(p(s(+(x64,x65))),y) |
|
→
|
+(s(p(+(x64,x65))),y) |
|
→
|
+(+(x64,x65),y) |
|
= |
t3
|
| t0
|
= |
+(p(+(s(x64),x65)),y) |
|
→
|
p(+(y,+(s(x64),x65))) |
|
→
|
p(+(y,s(+(x64,x65)))) |
|
= |
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.