Certification Problem

Input (TPDB TRS_Standard/Der95/06)

The rewrite relation of the following TRS is considered.

f(g(x))	→	g(g(f(x)))	(1)
f(g(x))	→	g(g(g(x)))	(2)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

f^#(g(x))

→

f^#(x)

(3)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(g(x)) → f^#(x) (3)

1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation

[f(x₁)] = 0

[f^#(x₁)] = x₁ + 0

[g(x₁)] = x₁ + 1

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(g(x)) → f^#(x) (3)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.