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 ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(h,x) → a^#(f,x) (4)

a^#(f,a(f,x)) → a^#(x,x) (3)

1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

a^#(h,x) → a^#(f,x) (4)

2 ≥ 2

a^#(f,a(f,x)) → a^#(x,x) (3)

2 > 2

2 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.