Certification Problem

Input (TPDB TRS_Standard/Mixed_TRS/motivation)

The rewrite relation of the following TRS is considered.

g(h(g(x)))	→	g(x)	(1)
g(g(x))	→	g(h(g(x)))	(2)
h(h(x))	→	h(f(h(x),x))	(3)

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.

g^#(g(x))	→	h^#(g(x))	(4)
h^#(h(x))	→	h^#(f(h(x),x))	(5)
g^#(g(x))	→	g^#(h(g(x)))	(6)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
g^#(g(x)) → g^#(h(g(x))) (6)

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

[h(x₁)] = 1

[f(x₁, x₂)] = x₂ + 1

[h^#(x₁)] = 0

[g^#(x₁)] = x₁ + 0

[g(x₁)] = x₁ + 2

together with the usable rule
h(h(x)) → h(f(h(x),x)) (3)
(w.r.t. the implicit argument filter of the reduction pair), the pair
g^#(g(x)) → g^#(h(g(x))) (6)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.