Certification Problem
Input (COPS 1140)
The rewrite relation of the following conditional TRS is considered.
|
f(y,a) |
→ |
g(x,x) |
| h(z) ≈ x, y ≈ k(z) |
|
k(x) |
→ |
h(x) |
|
k(x) |
→ |
b |
|
k(x) |
→ |
a |
| a |
→ |
b |
Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by ConCon @ CoCo 2020)
1 Inlining of Conditions
Inlining of conditions results in the following transformed CTRS having the
same multistep rewrite relation.
|
f(y,a) |
→ |
g(h(z),h(z)) |
| y ≈ k(z) |
|
k(x) |
→ |
h(x) |
|
k(x) |
→ |
b |
|
k(x) |
→ |
a |
| a |
→ |
b |
1.1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
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.