Certification Problem

Input (TPDB TRS_Standard/SK90/2.55)

The rewrite relation of the following TRS is considered.

f(x,g(x))	→	x	(1)
f(x,h(y))	→	f(h(x),y)	(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^#(x,h(y))

→

f^#(h(x),y)

(3)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(x,h(y)) → f^#(h(x),y) (3)

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

[h(x₁)] = x₁ + 1

[f(x₁, x₂)] = 0

[f^#(x₁, x₂)] = x₂ + 0

[g(x₁)] = 0

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

The dependency pairs are split into 0 components.