Certification Problem

Input (TPDB TRS_Standard/AG01/#3.31)

The rewrite relation of the following TRS is considered.

f(g(x))

→

f(a(g(g(f(x))),g(f(x))))

(1)

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)	(2)
f^#(g(x))	→	f^#(x)	(2)
f^#(g(x))	→	f^#(a(g(g(f(x))),g(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) (2)

f^#(g(x)) → f^#(x) (2)

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

[a(x₁, x₂)] = 0

[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 pairs

f^#(g(x)) → f^#(x) (2)

f^#(g(x)) → f^#(x) (2)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.