Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex23_Luc06_GM)

The rewrite relation of the following TRS is considered.

a__f(f(a))	→	c(f(g(f(a))))	(1)
mark(f(X))	→	a__f(mark(X))	(2)
mark(a)	→	a	(3)
mark(c(X))	→	c(X)	(4)
mark(g(X))	→	g(mark(X))	(5)
a__f(X)	→	f(X)	(6)

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)	(7)
mark^#(g(X))	→	mark^#(X)	(8)
mark^#(f(X))	→	a__f^#(mark(X))	(9)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(g(X)) → mark^#(X) (8)

mark^#(f(X)) → mark^#(X) (7)

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

[a] = 0

[a__f(x₁)] = 0

[c(x₁)] = 0

[f(x₁)] = x₁ + 1

[mark^#(x₁)] = x₁ + 0

[mark(x₁)] = 0

[a__f^#(x₁)] = 0

[g(x₁)] = x₁ + 1

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

mark^#(g(X)) → mark^#(X) (8)

mark^#(f(X)) → mark^#(X) (7)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.