Certification Problem

Input (TPDB TRS_Standard/Zantema_05/z04)

The rewrite relation of the following TRS is considered.

a(f,a(f,x))	→	a(x,x)	(1)
a(h,x)	→	a(f,a(g,a(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.

a^#(h,x)	→	a^#(g,a(f,x))	(3)
a^#(h,x)	→	a^#(f,x)	(4)
a^#(h,x)	→	a^#(f,a(g,a(f,x)))	(5)
a^#(f,a(f,x))	→	a^#(x,x)	(6)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(f,a(f,x)) → a^#(x,x) (6)

a^#(h,x) → a^#(f,x) (4)

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

[a(x₁, x₂)] = x₁ + x₂ + 21238

[h] = 1

[f] = 1

[a^#(x₁, x₂)] = x₂ + 0

[g] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
a^#(f,a(f,x)) → a^#(x,x) (6)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.