Certification Problem

Input (TPDB TRS_Innermost/AG01_innermost/#4.3)

The rewrite relation of the following TRS is considered.

f(g(x,y),x,z) → f(z,z,z) (1)
g(x,y) → x (2)
g(x,y) → y (3)
The evaluation strategy is innermost.

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
f#(g(x,y),x,z) → f#(z,z,z) (4)

1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

f#(g(x,y),x,z) → f#(z,z,z) (4)
3 ≥ 1
3 ≥ 2
3 ≥ 3

As there is no critical graph in the transitive closure, there are no infinite chains.