Certification Problem

Input (TPDB TRS_Standard/Der95/04)

The rewrite relation of the following TRS is considered.

f(f(x))	→	g(f(x))	(1)
g(g(x))	→	f(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))	→	g^#(f(x))	(3)
g^#(g(x))	→	f^#(x)	(4)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

g^#(g(x)) → f^#(x) (4)

f^#(f(x)) → g^#(f(x)) (3)

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

[f(x₁)] = x₁ + 2

[f^#(x₁)] = x₁ + 1

[g^#(x₁)] = x₁ + 0

[g(x₁)] = x₁ + 2

together with the usable rules

f(f(x)) → g(f(x)) (1)

g(g(x)) → f(x) (2)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

g^#(g(x)) → f^#(x) (4)

f^#(f(x)) → g^#(f(x)) (3)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.