Certification Problem

Input (TPDB TRS_Standard/Rubio_04/aoto)

The rewrite relation of the following TRS is considered.

f(f(X))	→	f(g(f(g(f(X)))))	(1)
f(g(f(X)))	→	f(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^#(f(X))	→	f^#(g(f(g(f(X)))))	(3)
f^#(f(X))	→	f^#(g(f(X)))	(4)
f^#(g(f(X)))	→	f^#(g(X))	(5)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(g(f(X))) → f^#(g(X)) (5)

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

[f(x₁)] = x₁ + 1

[f^#(x₁)] = x₁ + 0

[g(x₁)] = x₁ + 28958

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(g(f(X))) → f^#(g(X)) (5)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.