Certification Problem

Input (TPDB TRS_Standard/Rubio_04/test829)

The rewrite relation of the following TRS is considered.

f(c(X,s(Y)))	→	f(c(s(X),Y))	(1)
g(c(s(X),Y))	→	f(c(X,s(Y)))	(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^#(c(X,s(Y)))	→	f^#(c(s(X),Y))	(3)
g^#(c(s(X),Y))	→	f^#(c(X,s(Y)))	(4)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(c(X,s(Y))) → f^#(c(s(X),Y)) (3)

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

[s(x₁)] = x₁ + 1

[c(x₁, x₂)] = x₂ + 28958

[f(x₁)] = 0

[f^#(x₁)] = x₁ + 0

[g^#(x₁)] = 0

[g(x₁)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(c(X,s(Y))) → f^#(c(s(X),Y)) (3)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.