Certification Problem

Input (TPDB TRS_Standard/Zantema_05/z08)

The rewrite relation of the following TRS is considered.

f(a,f(b,f(a,x)))	→	f(a,f(b,f(b,f(a,x))))	(1)
f(b,f(b,f(b,x)))	→	f(b,f(b,x))	(2)

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

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

1

2

[b] =

1

1

[f(x₁, x₂)] =

0 0

0 1

· x₁ +

0 1

0 0

· x₂ +

8946

0

[f^#(x₁, x₂)] =

1 1

1 1

· x₂ +

0

0

together with the usable rules

f(a,f(b,f(a,x))) → f(a,f(b,f(b,f(a,x)))) (1)

f(b,f(b,f(b,x))) → f(b,f(b,x)) (2)

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

The dependency pairs are split into 0 components.