Certification Problem

Input (TPDB SRS_Standard/Gebhardt_06/06)

The rewrite relation of the following TRS is considered.

0(0(0(0(x1))))	→	0(1(0(1(x1))))	(1)
1(0(0(1(x1))))	→	0(1(0(0(x1))))	(2)

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by AProVE @ termCOMP 2023)

1 Looping derivation

There is a looping derivation.

0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ ε 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 ε

The derivation can be derived as follows.

0 0 0 0 →⁺ 0 1 0 1: This is an original rule (OC1).
1 0 0 1 →⁺ 0 1 0 0: This is an original rule (OC1).
0 0 0 0 0 0 1 →⁺ 0 1 0 0 1 0 0: The overlap closure is obtained from the following two overlap closures (OC2).
- 0 0 0 0 →⁺ 0 1 0 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 →⁺ 0 0 1 0 0 0 0: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 →⁺ 0 1 0 0 1 0 0
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 →⁺ 0 0 1 0 0 1 0 1: The overlap closure is obtained from the following two overlap closures (OC2).
- 0 0 0 0 0 0 1 →⁺ 0 0 1 0 0 0 0
- 0 0 0 0 →⁺ 0 1 0 1
0 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 0 0 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 →⁺ 0 0 1 0 0 1 0 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 0 0 0 0 0 1 0 0 0 1: The overlap closure is obtained from the following two overlap closures (OC2).
- 0 0 0 0 0 0 1 →⁺ 0 0 1 0 0 0 0
- 0 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 0 0 1
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 1 0 1 0 0 1 0 0 0 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 0 0 0 0 0 1 0 0 0 1
- 0 0 0 0 →⁺ 0 1 0 1
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 1 0 0 1 0 0 0 0 0 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 1 0 1 0 0 1 0 0 0 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 1 0 0 1 0 0 1 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 1 0 0 1 0 0 0 0 0 1
- 0 0 0 0 →⁺ 0 1 0 1
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 0 1 0 0 0 0 1 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 1 0 0 1 0 0 1 0 1 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 0 0 0 0 0 1 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 1 0 0 1 0 0 0 0 1 0 1 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 1 0 1 0 0 1 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 0 0 0 0 0 1 0 1 1
- 0 0 0 0 →⁺ 0 1 0 1
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 1 0 0 1 0 0 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 1 0 1 0 0 1 0 1 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 0 1 0 0 0 0 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 1 0 0 1 0 0 0 1 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 1 0 0 0 0 0 0 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 1 0 0 1 0 0 0 0 0 1 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 1 0 0 1 0 1 0 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 1 0 0 0 0 0 0 0 1 1
- 0 0 0 0 →⁺ 0 1 0 1
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 0 0 1 0 0 1 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 1 0 0 1 0 1 0 0 1 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 0 0 0 1 0 0 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 0 0 1 0 0 1 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 1 0 1 1 0 0 1: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 0 0 0 1 0 0 1
- 0 0 0 0 →⁺ 0 1 0 1
0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 1 0 1 0 1 0 0: The overlap closure is obtained from the following two overlap closures (OC2).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 1 0 1 1 0 0 1
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0: The overlap closure is obtained from the following two overlap closures (OC2).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 →⁺ 0 0 0 0 0 1 0 1 0 1 0 1 0 0
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0: The overlap closure is obtained from the following two overlap closures (OC3).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0
- 1 0 0 1 →⁺ 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1: The overlap closure is obtained from the following two overlap closures (OC2).
- 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 →⁺ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
- 0 0 0 0 →⁺ 0 1 0 1