Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw44)

The rewrite relation of the following TRS is considered.

f(a,f(f(a,x),a))

→

f(f(a,f(a,x)),a)

(1)

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^#(a,f(f(a,x),a))	→	f^#(a,f(a,x))	(2)
f^#(a,f(f(a,x),a))	→	f^#(f(a,f(a,x)),a)	(3)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(a,f(f(a,x),a)) → f^#(a,f(a,x)) (2)

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

[a] = 1

[f(x₁, x₂)] = x₁ + x₂ + 21238

[f^#(x₁, x₂)] = x₂ + 0

together with the usable rule
f(a,f(f(a,x),a)) → f(f(a,f(a,x)),a) (1)
(w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(a,f(f(a,x),a)) → f^#(a,f(a,x)) (2)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.