Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_15_Bor03_L)

The rewrite relation of the following TRS is considered.

f(0)	→	cons(0)	(1)
f(s(0))	→	f(p(s(0)))	(2)
p(s(0))	→	0	(3)

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(0))	→	p^#(s(0))	(4)
f^#(s(0))	→	f^#(p(s(0)))	(5)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(s(0)) → f^#(p(s(0))) (5)

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

[s(x₁)] = 2

[p^#(x₁)] = 0

[f(x₁)] = 0

[p(x₁)] = 1

[0] = 1

[f^#(x₁)] = x₁ + 0

[cons(x₁)] = 0

together with the usable rule
p(s(0)) → 0 (3)
(w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(s(0)) → f^#(p(s(0))) (5)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.