Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw16)

The rewrite relation of the following TRS is considered.

f(a,f(x,a))

→

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

1.1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[a] =

2

1

[f(x₁, x₂)] =

0 0

1 0

· x₁ +

0 0

0 1

· x₂ +

1

0

[f^#(x₁, x₂)] =

1 0

1 0

· x₁ +

0 1

1 1

· x₂ +

0

0

together with the usable rule
f(a,f(x,a)) → f(x,f(f(f(a,a),a),a)) (1)
(w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(a,f(x,a)) → f^#(x,f(f(f(a,a),a),a)) (4)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.