Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw43)

The rewrite relation of the following TRS is considered.

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

→

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

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

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

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

[a] = 40650

[f(x₁, x₂)] = max(x₁ + 4, x₂ + 1, 0)

[f^#(x₁, x₂)] = max(x₁ + 22250, x₂ + 22248, 0)

together with the usable rule
f(x,f(f(a,a),a)) → f(f(a,f(a,a)),x) (1)
(w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(x,f(f(a,a),a)) → f^#(f(a,f(a,a)),x) (3)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.