Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-249)

The rewrite relation of the following TRS is considered.

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

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 c c →⁺ b c b c a a b b c c ε

The derivation can be derived as follows.

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