Certification Problem

Input (TPDB TRS_Standard/Rubio_04/gmnp)

The rewrite relation of the following TRS is considered.

f(a)	→	f(c(a))	(1)
f(c(X))	→	X	(2)
f(c(a))	→	f(d(b))	(3)
f(a)	→	f(d(a))	(4)
f(d(X))	→	X	(5)
f(c(b))	→	f(d(a))	(6)
e(g(X))	→	e(X)	(7)

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.

e^#(g(X))	→	e^#(X)	(8)
f^#(a)	→	f^#(d(a))	(9)
f^#(c(a))	→	f^#(d(b))	(10)
f^#(c(b))	→	f^#(d(a))	(11)
f^#(a)	→	f^#(c(a))	(12)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
e^#(g(X)) → e^#(X) (8)

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

[a] = 0

[d(x₁)] = 0

[b] = 0

[e^#(x₁)] = x₁ + 0

[c(x₁)] = 0

[f(x₁)] = 0

[f^#(x₁)] = 0

[e(x₁)] = 0

[g(x₁)] = x₁ + 1

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
e^#(g(X)) → e^#(X) (8)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.