Certification Problem
Input (COPS 1155)
We consider the TRS containing the following rules:
|
plus(x,0) |
→ |
x |
(1) |
|
plus(0,1) |
→ |
1 |
(2) |
|
plus(x,plus(y,1)) |
→ |
plus(plus(x,y),1) |
(3) |
|
times(x,0) |
→ |
0 |
(4) |
|
times(x,1) |
→ |
x |
(5) |
|
times(x,plus(y,1)) |
→ |
plus(times(x,y),x) |
(6) |
|
m(0) |
→ |
0 |
(7) |
|
plus(m(1),1) |
→ |
0 |
(8) |
|
plus(m(plus(x,1)),1) |
→ |
m(x) |
(9) |
|
m(m(x)) |
→ |
x |
(10) |
|
plus(x,m(y)) |
→ |
m(plus(m(x),y)) |
(11) |
|
times(x,m(y)) |
→ |
m(times(x,y)) |
(12) |
The underlying signature is as follows:
{plus/2, 0/0, 1/0, times/2, m/1}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
|
= |
times(x,plus(0,1)) |
|
→
|
times(x,1) |
|
→
|
x |
|
= |
t2
|
| t0
|
= |
times(x,plus(0,1)) |
|
→
|
plus(times(x,0),x) |
|
→
|
plus(0,x) |
|
= |
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.