Certification Problem

Input (TPDB TRS_Standard/Mixed_TRS/test1)

The rewrite relation of the following TRS is considered.

f(s(x),y)	→	f(x,s(s(x)))	(1)
f(x,s(s(y)))	→	f(y,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^#(s(x),y)	→	f^#(x,s(s(x)))	(3)
f^#(x,s(s(y)))	→	f^#(y,x)	(4)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(x,s(s(y))) → f^#(y,x) (4)

f^#(s(x),y) → f^#(x,s(s(x))) (3)

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

[s(x₁)] = x₁ + 2

[f(x₁, x₂)] = max(0)

[f^#(x₁, x₂)] = max(x₁ + 4, x₂ + 1, 0)

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

f^#(x,s(s(y))) → f^#(y,x) (4)

f^#(s(x),y) → f^#(x,s(s(x))) (3)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.