Certification Problem

Input (TPDB TRS_Standard/Various_04/15)

The rewrite relation of the following TRS is considered.

f(s(x)) → s(s(f(p(s(x))))) (1)
f(0) → 0 (2)
p(s(x)) → x (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[f(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
1 0 0
[0] =
1 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 0
0 0 1
0 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[p(x1)] =
1 0 0
0 0 1
0 1 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
f(0) → 0 (2)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
f#(s(x)) → p#(s(x)) (4)
f#(s(x)) → f#(p(s(x))) (5)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.