Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw47)

The rewrite relation of the following TRS is considered.

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

→

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(f(a,f(x,a)),a) → f^#(f(x,a),a) (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)] = x₁ + 0

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

The dependency pairs are split into 0 components.