Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw40)

The rewrite relation of the following TRS is considered.

f(f(x,a),a)

→

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

(1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

f^#(f(x,a),a)	→	f^#(f(f(a,f(a,a)),a),x)	(2)
f^#(f(x,a),a)	→	f^#(f(a,f(a,a)),a)	(3)
f^#(f(x,a),a)	→	f^#(a,f(a,a))	(4)
f^#(f(x,a),a)	→	f^#(a,a)	(5)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

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

1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[f(x₁, x₂)] = 1 · x₁ + 1 · x₂

[a] = 0

[f^#(x₁, x₂)] = 1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[f^#(x₁, x₂)] = 2 · x₁ + 2 · x₂

[f(x₁, x₂)] = -1 + x₁

[a] = 2

the pair
f^#(f(x,a),a) → f^#(f(f(a,f(a,a)),a),x) (2)
could be deleted.
1.1.1.1.1 P is empty

There are no pairs anymore.