Certification Problem

Input (TPDB TRS_Standard/SK90/2.47)

The rewrite relation of the following TRS is considered.

a(b(x))	→	b(a(x))	(1)
a(c(x))	→	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.

a^#(b(x))

→

a^#(x)

(3)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
a^#(b(x)) → a^#(x) (3)

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

[a(x₁)] = 0

[b(x₁)] = x₁ + 1

[c(x₁)] = 0

[a^#(x₁)] = x₁ + 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
a^#(b(x)) → a^#(x) (3)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.