Certification Problem
Input (COPS 381)
The rewrite relation of the following conditional TRS is considered.
| a |
→ |
c |
| a |
→ |
d |
| b |
→ |
c |
| b |
→ |
d |
| c |
→ |
e |
| c |
→ |
k |
| d |
→ |
k |
|
f(x) |
→ |
x |
| x ≈ e
|
|
g(x,x) |
→ |
C |
| A ≈ B
|
|
h(x) |
→ |
i(x,x) |
Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by ConCon @ CoCo 2020)
1 Removal of Infeasible Rules
We may safely remove rules with infeasible conditions. They do not
influence the rewrite relation in any way.
1.1 Rules with Infeasible Conditions
-
1.1.1 Rule with Infeasible Conditions
The rule
has infeasible conditions.
1.1.1.1 Infeasible Equation
The equation
is infeasible.
1.1.1.1.1 Non-reachability
We show non-reachability w.r.t. the underlying TRS.
1.1.1.1.1.1 Non-reachability by TCAP
Non-reachability is shown by the TCAP approximation.
1.2 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.