Certification Problem
Input (COPS 319)
The rewrite relation of the following conditional TRS is considered.
|
f(x) |
→ |
e |
| d ≈ l
|
|
h(x,x) |
→ |
A |
Property / Task
Prove or disprove confluence.Answer / Result
Yes.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 Quasi-reductive SDTRS where all CCPs are joinable
The given strongly deterministic oriented 3-CTRS is quasi-reductive and all CCPs are joinable.
1.2.1 Quasi-Reductive CTRS
The given CTRS is quasi-reductive
1.2.1.1 Unraveling
To prove that the CTRS is quasi-reductive, we show termination of the following
unraveled system.
1.2.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(A) |
= |
0 |
|
weight(A) |
= |
2 |
|
|
|
| prec(h) |
= |
1 |
|
weight(h) |
= |
0 |
|
|
|
all of the following rules can be deleted.
1.2.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
1.2.2 All CCPs are joinable
A CCP is joinable if it is context-joinable, infeasible, or unfeasible.