Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-270)

The rewrite relation of the following TRS is considered.

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

The derivation can be derived as follows.

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