Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/27280)

The rewrite relation of the following TRS is considered.

0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(x1))))))))))	(1)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))	(2)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))	(3)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))	(4)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))	(5)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))	(6)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))	(7)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))))))	(8)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))))))	(9)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))))))))))))	(10)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))))))))))))	(11)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))))))))))))))))))	(12)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))))))))))))))))))	(13)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))	(14)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(15)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(16)
0(1(2(1(x1))))	→	1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(17)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

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

1(2(1(0(x1))))	→	2(1(0(2(1(0(1(1(2(1(x1))))))))))	(18)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))	(19)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))	(20)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))	(21)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))	(22)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))	(23)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))	(24)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))	(25)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))	(26)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))	(27)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))	(28)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))	(29)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))))))	(30)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))	(31)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(32)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(33)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(34)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

1^#(2(1(0(x1))))	→	1^#(x1)	(35)
1^#(2(1(0(x1))))	→	1^#(2(1(x1)))	(36)
1^#(2(1(0(x1))))	→	1^#(1(2(1(x1))))	(37)
1^#(2(1(0(x1))))	→	1^#(0(1(1(2(1(x1))))))	(38)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(1(1(2(1(x1)))))))))	(39)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))	(40)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))	(41)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))	(42)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))	(43)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))	(44)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))	(45)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))	(46)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))	(47)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))	(48)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))	(49)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))	(50)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))))	(51)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))))))))	(52)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(53)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(54)
1^#(2(1(0(x1))))	→	1^#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(55)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

1^#(2(1(0(x1))))	→	1^#(2(1(x1)))	(36)
1^#(2(1(0(x1))))	→	1^#(1(2(1(x1))))	(37)
1^#(2(1(0(x1))))	→	1^#(x1)	(35)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[2(x₁)]	=	-6 · x₁ + 0
[1^#(x₁)]	=	0 · x₁ + 0
[1(x₁)]	=	0 · x₁ + 0
[0(x₁)]	=	6 · x₁ + 7

together with the usable rules

1(2(1(0(x1))))	→	2(1(0(2(1(0(1(1(2(1(x1))))))))))	(18)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))	(19)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))	(20)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))	(21)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))	(22)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))	(23)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))	(24)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))	(25)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))	(26)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))	(27)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))	(28)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))	(29)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))))))	(30)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))	(31)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(32)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(33)
1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(34)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

1^#(2(1(0(x1))))	→	1^#(2(1(x1)))	(36)
1^#(2(1(0(x1))))	→	1^#(1(2(1(x1))))	(37)

could be deleted.

1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

1^#(2(1(0(x1))))	→	1^#(x1)	(35)

1	>	1

As there is no critical graph in the transitive closure, there are no infinite chains.