Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/ExConc_Zan97_GM)

The rewrite relation of the following TRS is considered.

a__f(X)	→	g(h(f(X)))	(1)
mark(f(X))	→	a__f(mark(X))	(2)
mark(g(X))	→	g(X)	(3)
mark(h(X))	→	h(mark(X))	(4)
a__f(X)	→	f(X)	(5)

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.

mark^#(f(X))	→	mark^#(X)	(6)
mark^#(h(X))	→	mark^#(X)	(7)
mark^#(f(X))	→	a__f^#(mark(X))	(8)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(h(X)) → mark^#(X) (7)

mark^#(f(X)) → mark^#(X) (6)

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

[h(x₁)] = x₁ + 1

[a__f(x₁)] = 0

[f(x₁)] = x₁ + 1

[mark^#(x₁)] = x₁ + 0

[mark(x₁)] = 0

[a__f^#(x₁)] = 0

[g(x₁)] = 0

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

mark^#(h(X)) → mark^#(X) (7)

mark^#(f(X)) → mark^#(X) (6)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.