Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z077)

The rewrite relation of the following TRS is considered.

f(0(x1))	→	s(0(x1))	(1)
d(0(x1))	→	0(x1)	(2)
d(s(x1))	→	s(s(d(x1)))	(3)
f(s(x1))	→	d(f(x1))	(4)

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^#(s(x1))	→	d^#(f(x1))	(5)
f^#(s(x1))	→	f^#(x1)	(6)
d^#(s(x1))	→	d^#(x1)	(7)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair
f^#(s(x1)) → f^#(x1) (6)

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

[d(x₁)] = 0

[s(x₁)] = x₁ + 1

[f(x₁)] = 0

[0(x₁)] = 0

[d^#(x₁)] = 0

[f^#(x₁)] = x₁ + 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(s(x1)) → f^#(x1) (6)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.
The 2^nd component contains the pair
d^#(s(x1)) → d^#(x1) (7)

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

[d(x₁)] = 0

[s(x₁)] = x₁ + 1

[f(x₁)] = 0

[0(x₁)] = 0

[d^#(x₁)] = x₁ + 0

[f^#(x₁)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
d^#(s(x1)) → d^#(x1) (7)
could be deleted.
1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.