Certification Problem

Input (TPDB TRS_Standard/AG01/#3.51)

The rewrite relation of the following TRS is considered.

f(f(x))	→	f(c(f(x)))	(1)
f(f(x))	→	f(d(f(x)))	(2)
g(c(x))	→	x	(3)
g(d(x))	→	x	(4)
g(c(h(0)))	→	g(d(1))	(5)
g(c(1))	→	g(d(h(0)))	(6)
g(h(x))	→	g(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.

g^#(c(1))	→	g^#(d(h(0)))	(8)
f^#(f(x))	→	f^#(c(f(x)))	(9)
g^#(h(x))	→	g^#(x)	(10)
f^#(f(x))	→	f^#(d(f(x)))	(11)
g^#(c(h(0)))	→	g^#(d(1))	(12)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
g^#(h(x)) → g^#(x) (10)

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

[h(x₁)] = x₁ + 1

[1] = 0

[d(x₁)] = 0

[c(x₁)] = 0

[f(x₁)] = 0

[0] = 0

[f^#(x₁)] = 0

[g^#(x₁)] = x₁ + 0

[g(x₁)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
g^#(h(x)) → g^#(x) (10)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.