Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-552)

The rewrite relation of the following TRS is considered.

a(b(x1))	→	x1	(1)
a(c(x1))	→	b(c(c(a(a(b(x1))))))	(2)
b(c(x1))	→	x1	(3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(c(x1)) → a^#(b(x1)) (6)

a^#(c(x1)) → a^#(a(b(x1))) (5)

1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the rationals with delta = 1/4

[a^#(x₁)] = 0 + 1/2 · x₁

[c(x₁)] = 1/2 + 4 · x₁

[b(x₁)] = 0 + 1/4 · x₁

[a(x₁)] = 0 + 4 · x₁

the pairs

a^#(c(x1)) → a^#(b(x1)) (6)

a^#(c(x1)) → a^#(a(b(x1))) (5)

could be deleted.
1.1.1.1 P is empty

There are no pairs anymore.