Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/259405)

The rewrite relation of the following TRS is considered.

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

1.1 Closure Under Flat Contexts

Using the flat contexts

{0(☐), 1(☐)}

We obtain the transformed TRS

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₁(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁ + 1
[1₀(x₁)]	=	1 · x₁ + 2
[0₀(x₁)]	=	1 · x₁

all of the following rules can be deleted.

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₁(x₁)]	=	1 · x₁ + 1
[1₁(x₁)]	=	1 · x₁ + 1
[1₀(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁ + 1

all of the following rules can be deleted.

0₁(1₁(1₁(1₀(0₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(1₁(1₁(1₁(1₁(1₁(1₀(0₀(0₁(1₁(1₀(0₁(1₀(0₁(1₁(1₁(1₀(0₀(0₀(0₀(0₀(0₀(0₁(1₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(1₁(1₁(1₁(1₁(1₁(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₀(0₀(0₁(1₁(1₀(0₀(0₁(1₁(1₀(0₀(0₀(0₀(0₁(1₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₀(0₀(0₀(0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(83)
0₁(1₁(1₁(1₀(0₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(1₁(1₁(1₁(1₁(1₁(1₀(0₀(0₁(1₁(1₀(0₁(1₀(0₁(1₁(1₁(1₀(0₀(0₀(0₀(0₀(0₀(0₁(1₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(1₁(1₁(1₁(1₁(1₁(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₀(0₀(0₁(1₁(1₀(0₀(0₁(1₁(1₀(0₀(0₀(0₀(0₁(1₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₀(0₀(0₀(0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(84)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₀(x₁)]	=	1 · x₁ + 1
[0₁(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁ + 1
[1₀(x₁)]	=	1 · x₁

all of the following rules can be deleted.

0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₁(1₁(1₁(1₀(0₁(1₁(1₀(0₀(0₀(0₀(0₀(0₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₀(0₀(0₀(0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(1₁(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0₁(1₀(0₀(0₀(0₁(1₀(0₀(0₁(1₁(1₁(1₁(1₀(0₁(1₁(1₀(0₀(0₀(0₀(0₀(0₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₀(0₀(0₀(0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(141)
0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₁(1₁(1₁(1₀(0₁(1₁(1₀(0₀(0₀(0₀(0₀(0₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₀(0₀(0₀(0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(1₁(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0₁(1₀(0₀(0₀(0₁(1₀(0₀(0₁(1₁(1₁(1₁(1₀(0₁(1₁(1₀(0₀(0₀(0₀(0₀(0₁(1₀(0₀(0₁(1₀(0₀(0₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₁(1₀(0₁(1₀(0₀(0₀(0₀(0₁(1₁(1₁(1₀(0₀(0₁(1₀(0₀(0₀(0₁(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(142)

1.1.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.

0(0(1(0(1(1(0(0(0(0(1(0(1(1(0(1(0(0(0(1(1(1(1(0(1(0(1(0(0(1(1(1(1(1(1(0(0(0(1(0(0(1(0(0(1(1(0(0(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(1)
0(0(1(0(1(1(0(0(0(0(1(0(1(1(0(1(0(0(0(1(1(1(1(0(1(0(1(0(0(1(1(1(1(1(1(0(0(0(1(0(0(1(0(0(1(1(0(0(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(0(0(0(0(0(0(1(1(1(1(1(0(1(0(0(1(1(0(1(0(0(0(1(0(0(1(1(1(0(0(0(1(0(1(0(1(1(0(0(1(0(0(0(1(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(2)
0(0(1(1(0(0(1(0(0(1(1(1(1(0(1(0(1(1(1(0(1(0(1(0(1(0(1(0(0(1(1(0(0(1(0(0(0(0(0(0(0(1(1(0(1(0(1(1(0(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(1(0(1(1(1(1(1(0(1(0(1(0(0(0(0(0(0(0(1(0(1(0(0(1(0(1(1(0(1(0(1(1(0(0(1(1(0(1(0(0(0(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(3)
0(0(1(1(0(0(1(0(0(1(1(1(1(0(1(0(1(1(1(0(1(0(1(0(1(0(1(0(0(1(1(0(0(1(0(0(0(0(0(0(0(1(1(0(1(0(1(1(0(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(1(0(0(1(1(0(1(0(1(0(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(4)
0(0(1(1(0(0(1(0(0(1(1(1(1(0(1(0(1(1(1(0(1(0(1(0(1(0(1(0(0(1(1(0(0(1(0(0(0(0(0(0(0(1(1(0(1(0(1(1(0(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(0(1(0(1(1(0(0(0(0(1(0(1(1(0(1(0(0(0(1(1(1(1(0(1(0(1(0(0(1(1(1(1(1(1(0(0(0(1(0(0(1(0(0(1(1(0(0(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(5)
0(1(0(0(1(1(0(0(0(0(1(1(1(0(1(1(0(0(0(1(1(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(0(1(1(0(1(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(1(0(0(1(1(0(1(0(1(0(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(6)
0(1(0(0(1(1(0(0(0(0(1(1(1(0(1(1(0(0(0(1(1(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(0(1(1(0(1(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(0(0(0(0(0(0(1(1(1(1(1(0(1(0(0(1(1(0(1(0(0(0(1(0(0(1(1(1(0(0(0(1(0(1(0(1(1(0(0(1(0(0(0(1(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(7)
0(1(0(0(1(1(0(0(0(0(1(1(1(0(1(1(0(0(0(1(1(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(0(1(1(0(1(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(0(1(0(1(1(0(0(0(0(1(0(1(1(0(1(0(0(0(1(1(1(1(0(1(0(1(0(0(1(1(1(1(1(1(0(0(0(1(0(0(1(0(0(1(1(0(0(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(8)
0(1(0(0(1(1(0(0(0(0(1(1(1(0(1(1(0(0(0(1(1(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(0(1(1(0(1(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(0(1(1(0(0(0(0(0(0(1(0(0(1(1(1(1(1(0(1(0(0(0(0(0(1(1(1(0(1(0(1(0(0(0(0(0(0(0(1(0(1(1(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(9)
0(1(0(0(1(1(0(0(0(0(1(1(1(0(1(1(0(0(0(1(1(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(0(1(1(0(1(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(1(0(0(0(0(0(1(1(0(1(0(0(1(0(0(0(0(1(1(0(0(1(0(0(1(1(0(0(1(1(1(0(1(0(1(1(1(0(1(0(0(1(0(1(0(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(10)
0(1(0(0(1(1(0(0(0(0(1(1(1(0(1(1(0(0(0(1(1(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(0(1(1(0(1(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(0(1(1(0(0(0(0(0(0(1(0(0(1(1(1(1(1(0(1(0(0(0(0(0(1(1(1(0(1(0(1(0(0(0(0(0(0(0(1(0(1(1(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(11)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(0(1(1(0(0(0(0(0(0(1(0(0(1(1(1(1(1(0(1(0(0(0(0(0(1(1(1(0(1(0(1(0(0(0(0(0(0(0(1(0(1(1(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(12)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(1(0(0(1(1(0(1(0(1(0(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(13)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(14)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(1(0(0(0(0(0(1(1(0(1(0(0(1(0(0(0(0(1(1(0(0(1(0(0(1(1(0(0(1(1(1(0(1(0(1(1(1(0(1(0(0(1(0(1(0(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(15)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(0(1(0(1(1(0(0(0(0(1(0(1(1(0(1(0(0(0(1(1(1(1(0(1(0(1(0(0(1(1(1(1(1(1(0(0(0(1(0(0(1(0(0(1(1(0(0(0(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(16)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(17)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(1(0(0(1(1(0(1(0(1(0(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(18)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(1(0(0(0(0(0(1(1(0(1(0(0(1(0(0(0(0(1(1(0(0(1(0(0(1(1(0(0(1(1(1(0(1(0(1(1(1(0(1(0(0(1(0(1(0(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(19)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(0(0(1(0(1(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(20)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(0(0(1(0(0(0(1(0(0(1(1(1(0(0(0(1(1(0(1(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(1(0(0(1(1(0(1(0(1(0(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(21)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(0(0(1(0(0(0(1(0(0(1(1(1(0(0(0(1(1(0(1(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	1(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(1(0(0(1(1(0(1(0(1(0(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(22)
0(1(0(0(1(1(0(0(1(1(0(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(0(0(1(0(0(0(1(0(0(1(1(1(0(0(0(1(1(0(1(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	→	0(0(0(0(0(0(0(1(1(1(1(1(0(1(0(0(1(1(0(1(0(0(0(1(0(0(1(1(1(0(0(0(1(0(1(0(1(1(0(0(1(0(0(0(1(0(0(1(1(0(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(23)
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