Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-227)

The rewrite relation of the following TRS is considered.

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

The derivation can be derived as follows.

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