Certification Problem

Input (TPDB TRS_Standard/AG01/#3.53a)

The rewrite relation of the following TRS is considered.

g(x,y)	→	x	(1)
g(x,y)	→	y	(2)
f(s(x),y,y)	→	f(y,x,s(x))	(3)

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,y)

→

f^#(y,x,s(x))

(4)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(s(x),y,y) → f^#(y,x,s(x)) (4)

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

[s(x₁)] = x₁ + 1

[f(x₁, x₂, x₃)] = 0

[f^#(x₁, x₂, x₃)] = x₁ + x₂ + 0

[g^#(x₁, x₂)] = 0

[g(x₁, x₂)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(s(x),y,y) → f^#(y,x,s(x)) (4)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.