Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-243)

The rewrite relation of the following TRS is considered.

a(x1)	→	x1	(1)
a(b(x1))	→	b(c(a(a(x1))))	(2)
c(b(x1))	→	x1	(3)
c(c(x1))	→	b(x1)	(4)

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by AProVE @ termCOMP 2023)

1 Looping derivation

There is a looping derivation.

a b b b b →⁺ b c a b b b b a

The derivation can be derived as follows.

a b →⁺ b c a a: This is an original rule (OC1).
a b b →⁺ b c a b c a a: The overlap closure is obtained from the following two overlap closures (OC2).
- a b →⁺ b c a a
- a b →⁺ b c a a
a →⁺ ε: This is an original rule (OC1).
a b →⁺ b c a: The overlap closure is obtained from the following two overlap closures (OC2).
- a b →⁺ b c a a
- a →⁺ ε
a b b →⁺ b c b c a c a a: The overlap closure is obtained from the following two overlap closures (OC3).
- a b b →⁺ b c a b c a a
- a b →⁺ b c a
a b b b →⁺ b c b c a c a b c a: The overlap closure is obtained from the following two overlap closures (OC2).
- a b b →⁺ b c b c a c a a
- a b →⁺ b c a
a b b b →⁺ b c b c c a b c a: The overlap closure is obtained from the following two overlap closures (OC3).
- a b b b →⁺ b c b c a c a b c a
- a →⁺ ε
c b →⁺ ε: This is an original rule (OC1).
a b b b →⁺ b c c a b c a: The overlap closure is obtained from the following two overlap closures (OC3).
- a b b b →⁺ b c b c c a b c a
- c b →⁺ ε
a b b b →⁺ b c c b c a c a: The overlap closure is obtained from the following two overlap closures (OC3).
- a b b b →⁺ b c c a b c a
- a b →⁺ b c a
c c →⁺ b: This is an original rule (OC1).
a b b b →⁺ b b b c a c a: The overlap closure is obtained from the following two overlap closures (OC3).
- a b b b →⁺ b c c b c a c a
- c c →⁺ b
a b b b b →⁺ b c a b b b c a c a: The overlap closure is obtained from the following two overlap closures (OC2).
- a b →⁺ b c a a
- a b b b →⁺ b b b c a c a
a b b b b →⁺ b c a b b b c c a: The overlap closure is obtained from the following two overlap closures (OC3).
- a b b b b →⁺ b c a b b b c a c a
- a →⁺ ε
a b b b b →⁺ b c a b b b b a: The overlap closure is obtained from the following two overlap closures (OC3).
- a b b b b →⁺ b c a b b b c c a
- c c →⁺ b