Certification Problem

Input (TPDB TRS_Innermost/AG01_innermost/#4.5)

The rewrite relation of the following TRS is considered.

f(0) → f(0) (1)
0 → 1 (2)
The evaluation strategy is innermost.

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals
[f(x1)] = 1 · x1
[0] = 1
[1] = 0
all of the following rules can be deleted.
0 → 1 (2)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
f#(0) → f#(0) (3)

1.1.1 Usable Rules Processor

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

There are no rules.

1.1.1.1 Size-Change Termination

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

f#(0) → f#(0) (3)
1 ≥ 1

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