Certification Problem

Input (TPDB TRS_Standard/Rubio_04/nestrec)

The rewrite relation of the following TRS is considered.

f(g(X))	→	g(f(f(X)))	(1)
f(h(X))	→	h(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)
f^#(g(X))	→	f^#(f(X))	(4)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(g(X)) → f^#(f(X)) (4)

f^#(g(X)) → f^#(X) (3)

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

[h(x₁)] = 1

[f(x₁)] = x₁ + 0

[f^#(x₁)] = x₁ + 0

[g(x₁)] = x₁ + 1

together with the usable rules

f(g(X)) → g(f(f(X))) (1)

f(h(X)) → h(g(X)) (2)

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

f^#(g(X)) → f^#(f(X)) (4)

f^#(g(X)) → f^#(X) (3)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.