Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z014)

The rewrite relation of the following TRS is considered.

a(b(x1))	→	b(a(a(x1)))	(1)
b(x1)	→	c(a(c(x1)))	(2)
a(a(x1))	→	a(c(a(x1)))	(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.

a^#(a(x1))	→	a^#(c(a(x1)))	(4)
a^#(b(x1))	→	a^#(x1)	(5)
a^#(b(x1))	→	b^#(a(a(x1)))	(6)
b^#(x1)	→	a^#(c(x1))	(7)
a^#(b(x1))	→	a^#(a(x1))	(8)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(x1)) → a^#(a(x1)) (8)

a^#(b(x1)) → a^#(x1) (5)

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

[a(x₁)] = x₁ + 0

[b(x₁)] = x₁ + 1

[c(x₁)] = 0

[a^#(x₁)] = x₁ + 0

[b^#(x₁)] = 0

together with the usable rules

a(b(x1)) → b(a(a(x1))) (1)

a(a(x1)) → a(c(a(x1))) (3)

b(x1) → c(a(c(x1))) (2)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

a^#(b(x1)) → a^#(a(x1)) (8)

a^#(b(x1)) → a^#(x1) (5)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.