Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/96642)

The rewrite relation of the following TRS is considered.

0(1(2(1(2(2(x1))))))	→	0(3(4(3(5(2(x1))))))	(1)
4(0(1(0(3(0(x1))))))	→	2(5(0(5(5(0(x1))))))	(2)
2(1(3(0(3(3(5(x1)))))))	→	2(5(2(2(5(5(2(x1)))))))	(3)
3(3(0(0(0(5(2(x1)))))))	→	5(3(3(1(2(5(2(x1)))))))	(4)
3(3(3(0(1(1(0(x1)))))))	→	5(3(3(0(2(1(0(x1)))))))	(5)
4(0(2(3(1(4(1(x1)))))))	→	1(4(3(2(5(4(x1))))))	(6)
4(1(5(0(0(5(2(1(3(x1)))))))))	→	4(5(0(3(5(2(2(2(3(x1)))))))))	(7)
5(2(5(1(5(0(3(0(4(2(x1))))))))))	→	3(3(2(5(1(3(3(5(0(2(x1))))))))))	(8)
2(1(4(2(2(0(3(0(1(0(5(x1)))))))))))	→	2(2(5(4(2(4(1(4(3(1(x1))))))))))	(9)
3(3(0(1(3(4(2(1(1(2(1(x1)))))))))))	→	5(2(4(4(3(5(3(5(5(1(1(x1)))))))))))	(10)
5(3(2(3(5(0(4(0(2(3(4(5(x1))))))))))))	→	3(5(2(1(1(5(2(1(0(5(4(x1)))))))))))	(11)
5(2(3(5(5(0(0(0(3(4(0(2(5(x1)))))))))))))	→	5(5(5(2(4(3(5(1(1(2(5(0(x1))))))))))))	(12)
5(5(3(4(3(2(4(0(5(2(1(2(2(x1)))))))))))))	→	5(5(4(4(0(1(0(1(3(1(4(2(x1))))))))))))	(13)
1(4(5(1(5(0(1(1(1(0(3(2(3(3(1(x1)))))))))))))))	→	1(1(3(2(3(4(3(5(4(4(3(5(3(1(x1))))))))))))))	(14)
3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1)))))))))))))))	→	3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1)))))))))))))))	(15)
3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1))))))))))))))))	→	3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1))))))))))))))))	(16)
5(0(2(5(5(1(3(2(5(2(0(4(4(0(4(1(x1))))))))))))))))	→	5(1(1(0(0(2(2(2(1(3(4(3(2(5(4(x1)))))))))))))))	(17)
3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1))))))))))))))))))	→	3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1))))))))))))))))))	(18)
4(5(3(3(2(5(5(1(3(5(1(5(0(0(2(5(5(1(x1))))))))))))))))))	→	4(5(5(2(2(3(0(3(3(0(4(1(3(1(1(1(x1))))))))))))))))	(19)
0(0(5(3(2(0(5(1(3(3(5(2(0(2(5(3(4(4(5(x1)))))))))))))))))))	→	0(0(5(3(5(5(0(1(3(5(0(1(3(1(3(4(2(4(x1))))))))))))))))))	(20)
2(5(1(2(2(3(2(3(2(1(4(2(0(5(5(1(0(3(3(x1)))))))))))))))))))	→	2(3(0(4(5(0(4(3(1(3(2(0(4(1(2(4(1(3(x1))))))))))))))))))	(21)
3(2(1(3(3(1(5(1(2(0(2(0(4(1(4(2(5(5(1(5(x1))))))))))))))))))))	→	5(1(0(1(3(1(1(4(2(0(0(2(3(2(5(3(5(1(2(x1)))))))))))))))))))	(22)
3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1))))))))))))))))))))	(23)
0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))))))))))	(24)
1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))))))))	(25)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0(x₁)]	=	1 · x₁ + 1
[1(x₁)]	=	1 · x₁ + 1
[2(x₁)]	=	1 · x₁ + 1
[3(x₁)]	=	1 · x₁ + 1
[4(x₁)]	=	1 · x₁ + 1
[5(x₁)]	=	1 · x₁ + 1

all of the following rules can be deleted.

4(0(2(3(1(4(1(x1)))))))	→	1(4(3(2(5(4(x1))))))	(6)
2(1(4(2(2(0(3(0(1(0(5(x1)))))))))))	→	2(2(5(4(2(4(1(4(3(1(x1))))))))))	(9)
5(3(2(3(5(0(4(0(2(3(4(5(x1))))))))))))	→	3(5(2(1(1(5(2(1(0(5(4(x1)))))))))))	(11)
5(2(3(5(5(0(0(0(3(4(0(2(5(x1)))))))))))))	→	5(5(5(2(4(3(5(1(1(2(5(0(x1))))))))))))	(12)
5(5(3(4(3(2(4(0(5(2(1(2(2(x1)))))))))))))	→	5(5(4(4(0(1(0(1(3(1(4(2(x1))))))))))))	(13)
1(4(5(1(5(0(1(1(1(0(3(2(3(3(1(x1)))))))))))))))	→	1(1(3(2(3(4(3(5(4(4(3(5(3(1(x1))))))))))))))	(14)
5(0(2(5(5(1(3(2(5(2(0(4(4(0(4(1(x1))))))))))))))))	→	5(1(1(0(0(2(2(2(1(3(4(3(2(5(4(x1)))))))))))))))	(17)
4(5(3(3(2(5(5(1(3(5(1(5(0(0(2(5(5(1(x1))))))))))))))))))	→	4(5(5(2(2(3(0(3(3(0(4(1(3(1(1(1(x1))))))))))))))))	(19)
0(0(5(3(2(0(5(1(3(3(5(2(0(2(5(3(4(4(5(x1)))))))))))))))))))	→	0(0(5(3(5(5(0(1(3(5(0(1(3(1(3(4(2(4(x1))))))))))))))))))	(20)
2(5(1(2(2(3(2(3(2(1(4(2(0(5(5(1(0(3(3(x1)))))))))))))))))))	→	2(3(0(4(5(0(4(3(1(3(2(0(4(1(2(4(1(3(x1))))))))))))))))))	(21)
3(2(1(3(3(1(5(1(2(0(2(0(4(1(4(2(5(5(1(5(x1))))))))))))))))))))	→	5(1(0(1(3(1(1(4(2(0(0(2(3(2(5(3(5(1(2(x1)))))))))))))))))))	(22)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 161 ruless (increase limit for explicit display).

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(2(1(2(2(x1))))))	→	3^#(4(3(5(2(x1)))))	(27)
3^#(3(0(0(0(5(2(x1)))))))	→	3^#(3(1(2(5(2(x1))))))	(44)
3^#(3(0(0(0(5(2(x1)))))))	→	3^#(1(2(5(2(x1)))))	(45)
3^#(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1))))))))))))))))	→	3^#(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1))))))))))))))))	(93)
3^#(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1))))))))))))))))	→	3^#(5(4(5(3(1(1(4(2(2(3(x1)))))))))))	(98)
3^#(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1))))))))))))))))	→	3^#(1(1(4(2(2(3(x1)))))))	(102)
3^#(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1))))))))))))))))	→	3^#(x1)	(108)
3^#(3(3(0(1(1(0(x1)))))))	→	3^#(3(0(2(1(0(x1))))))	(49)
3^#(3(3(0(1(1(0(x1)))))))	→	3^#(0(2(1(0(x1)))))	(50)
3^#(3(3(0(1(1(0(x1)))))))	→	2^#(1(0(x1)))	(52)
2^#(1(3(0(3(3(5(x1)))))))	→	5^#(2(x1))	(41)
5^#(2(5(1(5(0(3(0(4(2(x1))))))))))	→	3^#(3(2(5(1(3(3(5(0(2(x1))))))))))	(61)
5^#(2(5(1(5(0(3(0(4(2(x1))))))))))	→	3^#(2(5(1(3(3(5(0(2(x1)))))))))	(62)
5^#(2(5(1(5(0(3(0(4(2(x1))))))))))	→	3^#(3(5(0(2(x1)))))	(66)
5^#(2(5(1(5(0(3(0(4(2(x1))))))))))	→	3^#(5(0(2(x1))))	(67)
2^#(1(3(0(3(3(5(x1)))))))	→	2^#(x1)	(42)
3^#(3(0(1(3(4(2(1(1(2(1(x1)))))))))))	→	3^#(5(3(5(5(1(1(x1)))))))	(74)
3^#(3(0(1(3(4(2(1(1(2(1(x1)))))))))))	→	3^#(5(5(1(1(x1)))))	(76)
3^#(3(0(1(3(4(2(1(1(2(1(x1)))))))))))	→	1^#(1(x1))	(79)
1^#(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	3^#(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))))))))	(166)
1^#(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	3^#(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))	(167)
1^#(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	3^#(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))))))	(168)
1^#(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	2^#(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))	(173)
1^#(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	3^#(5(3(4(4(0(x1))))))	(181)
1^#(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	3^#(4(4(0(x1))))	(183)
1^#(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1)))))))))))))))))))))	→	0^#(x1)	(186)
0^#(1(2(1(2(2(x1))))))	→	3^#(5(2(x1)))	(29)
0^#(1(2(1(2(2(x1))))))	→	5^#(2(x1))	(30)
0^#(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	2^#(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))))))))))	(147)
0^#(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	3^#(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))))))	(151)
0^#(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	3^#(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))	(154)
0^#(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	3^#(1(4(2(1(5(3(3(5(5(0(x1)))))))))))	(157)
0^#(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	2^#(1(5(3(3(5(5(0(x1))))))))	(160)
0^#(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	3^#(3(5(5(0(x1)))))	(163)
0^#(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1)))))))))))))))))))))	→	3^#(5(5(0(x1))))	(164)
3^#(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1)))))))))))))))	→	3^#(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1)))))))))))))))	(80)
3^#(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1)))))))))))))))	→	2^#(1(1(3(3(0(2(2(1(2(0(3(3(x1)))))))))))))	(82)
3^#(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1)))))))))))))))	→	3^#(3(0(2(2(1(2(0(3(3(x1))))))))))	(85)
3^#(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1)))))))))))))))	→	3^#(0(2(2(1(2(0(3(3(x1)))))))))	(86)
3^#(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1)))))))))))))))	→	2^#(1(2(0(3(3(x1))))))	(89)
3^#(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1))))))))))))))))))	→	3^#(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1))))))))))))))))))	(109)
3^#(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1))))))))))))))))))	→	0^#(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))	(114)
3^#(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1))))))))))))))))))	→	3^#(4(5(0(2(3(1(x1)))))))	(120)
3^#(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1))))))))))))))))))	→	3^#(1(x1))	(125)
3^#(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1))))))))))))))))))	→	1^#(x1)	(126)
3^#(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	3^#(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1))))))))))))))))))))	(127)
3^#(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	2^#(1(0(1(5(0(3(3(5(2(x1))))))))))	(137)
3^#(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	0^#(1(5(0(3(3(5(2(x1))))))))	(139)
3^#(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	3^#(3(5(2(x1))))	(143)
3^#(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	3^#(5(2(x1)))	(144)
3^#(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	5^#(2(x1))	(145)
3^#(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1))))))))))))))))))))	→	2^#(x1)	(146)

1.1.1.1 Reduction Pair Processor