Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213865)

The rewrite relation of the following TRS is considered.

0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 227 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

There are 109 ruless (increase limit for explicit display).

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	x₁ + 13508
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 24389
[3(x₁)]	=	x₁ + 0
[4^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	x₁ + 1
[5^#(x₁)]	=	x₁ + 24390
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	x₁ + 13507

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

5^#(4(3(4(1(x1)))))	→	4^#(4(1(5(x1))))	(165)
0^#(5(2(3(1(x1)))))	→	1^#(5(4(x1)))	(271)
3^#(0(2(5(1(x1)))))	→	1^#(4(x1))	(270)
5^#(0(1(1(4(x1)))))	→	1^#(5(1(4(x1))))	(162)
3^#(0(1(1(x1))))	→	4^#(1(1(0(x1))))	(160)
1^#(4(0(1(2(x1)))))	→	3^#(1(x1))	(269)
1^#(4(0(1(2(x1)))))	→	1^#(x1)	(265)
5^#(4(3(4(1(x1)))))	→	5^#(x1)	(152)
0^#(5(2(3(1(x1)))))	→	4^#(x1)	(151)
3^#(3(1(1(2(x1)))))	→	1^#(4(3(x1)))	(262)
4^#(0(1(1(4(x1)))))	→	1^#(0(3(4(4(1(x1))))))	(148)
3^#(3(1(1(1(x1)))))	→	3^#(1(1(x1)))	(145)
4^#(5(3(2(1(x1)))))	→	4^#(x1)	(144)
3^#(0(2(5(1(x1)))))	→	0^#(1(x1))	(143)
3^#(0(1(0(2(x1)))))	→	4^#(1(x1))	(256)
0^#(3(3(2(1(x1)))))	→	4^#(x1)	(252)
5^#(0(2(3(1(x1)))))	→	0^#(x1)	(253)
3^#(3(1(1(4(x1)))))	→	1^#(x1)	(250)
5^#(3(1(1(4(x1)))))	→	1^#(x1)	(136)
3^#(0(1(1(x1))))	→	1^#(0(x1))	(249)
0^#(5(2(3(1(x1)))))	→	5^#(4(x1))	(133)
0^#(3(3(2(1(x1)))))	→	1^#(4(x1))	(244)
3^#(0(2(1(x1))))	→	1^#(4(x1))	(129)
3^#(3(1(1(2(x1)))))	→	4^#(3(x1))	(242)
3^#(3(1(1(4(x1)))))	→	4^#(4(3(1(3(1(x1))))))	(125)
5^#(4(3(4(1(x1)))))	→	4^#(1(5(x1)))	(126)
3^#(5(0(2(1(x1)))))	→	4^#(1(x1))	(239)
4^#(0(1(1(4(x1)))))	→	0^#(3(4(4(1(x1)))))	(121)
3^#(0(2(1(x1))))	→	0^#(x1)	(71)
3^#(0(2(5(1(x1)))))	→	4^#(x1)	(232)
5^#(0(1(1(4(x1)))))	→	5^#(1(4(x1)))	(233)
3^#(0(2(1(x1))))	→	4^#(x1)	(230)
1^#(4(0(1(2(x1)))))	→	4^#(0(3(1(x1))))	(227)
3^#(0(4(5(1(x1)))))	→	0^#(0(3(4(1(x1)))))	(228)
0^#(2(3(1(x1))))	→	1^#(4(x1))	(223)
0^#(1(0(1(2(x1)))))	→	1^#(1(3(x1)))	(109)
0^#(1(5(3(1(x1)))))	→	1^#(5(4(1(x1))))	(218)
3^#(3(1(1(2(x1)))))	→	3^#(x1)	(219)
5^#(3(1(1(2(x1)))))	→	5^#(x1)	(108)
3^#(0(2(1(x1))))	→	1^#(0(x1))	(106)
3^#(0(1(1(x1))))	→	0^#(x1)	(105)
0^#(5(4(3(1(x1)))))	→	5^#(x1)	(215)
3^#(3(1(1(4(x1)))))	→	1^#(3(1(x1)))	(102)
5^#(4(3(4(1(x1)))))	→	1^#(5(x1))	(103)
5^#(3(0(1(1(x1)))))	→	0^#(x1)	(212)
4^#(0(1(1(4(x1)))))	→	4^#(1(x1))	(213)
4^#(0(1(1(4(x1)))))	→	4^#(4(1(x1)))	(101)
1^#(4(0(1(2(x1)))))	→	0^#(3(1(x1)))	(211)
0^#(2(3(1(x1))))	→	4^#(x1)	(208)
0^#(5(4(3(1(x1)))))	→	4^#(5(x1))	(95)
3^#(3(1(1(4(x1)))))	→	3^#(1(x1))	(204)
4^#(0(1(1(4(x1)))))	→	1^#(x1)	(91)
5^#(3(1(1(2(x1)))))	→	1^#(4(5(x1)))	(92)
5^#(3(0(2(1(x1)))))	→	5^#(x1)	(202)
4^#(0(1(1(4(x1)))))	→	3^#(4(4(1(x1))))	(87)
3^#(0(4(5(1(x1)))))	→	3^#(4(1(x1)))	(83)
3^#(0(1(0(2(x1)))))	→	1^#(x1)	(197)
5^#(3(0(1(1(x1)))))	→	1^#(1(5(0(x1))))	(78)
3^#(0(1(2(x1))))	→	4^#(x1)	(196)
5^#(3(0(2(1(x1)))))	→	4^#(x1)	(195)
0^#(1(5(3(1(x1)))))	→	5^#(4(1(x1)))	(76)
3^#(0(1(1(x1))))	→	1^#(1(0(x1)))	(75)
5^#(3(0(1(1(x1)))))	→	1^#(5(0(x1)))	(191)
3^#(3(1(1(4(x1)))))	→	4^#(3(1(3(1(x1)))))	(74)
3^#(0(2(1(x1))))	→	0^#(x1)	(71)
0^#(1(0(1(2(x1)))))	→	1^#(3(x1))	(70)
0^#(2(4(3(1(x1)))))	→	0^#(x1)	(186)
3^#(3(1(1(1(x1)))))	→	1^#(3(1(1(x1))))	(68)
0^#(5(4(3(1(x1)))))	→	4^#(1(4(5(x1))))	(66)
3^#(0(2(1(x1))))	→	1^#(0(x1))	(106)
5^#(3(0(1(1(x1)))))	→	5^#(0(x1))	(62)
0^#(5(4(3(1(x1)))))	→	1^#(4(5(x1)))	(60)
0^#(1(0(1(2(x1)))))	→	3^#(x1)	(179)
3^#(0(4(5(1(x1)))))	→	0^#(3(4(1(x1))))	(57)
5^#(3(1(1(2(x1)))))	→	4^#(5(x1))	(174)
5^#(0(2(3(1(x1)))))	→	1^#(0(x1))	(172)
3^#(4(3(2(1(x1)))))	→	4^#(1(x1))	(54)
0^#(1(5(3(1(x1)))))	→	4^#(1(x1))	(170)
3^#(0(4(5(1(x1)))))	→	4^#(1(x1))	(168)
5^#(3(0(2(1(x1)))))	→	1^#(5(x1))	(51)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

1^#(4(0(1(2(x1)))))

→

1^#(4(0(3(1(x1)))))

(156)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	0
[3(x₁)]	=	0
[4^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	x₁ + 1
[5^#(x₁)]	=	x₁ + 24390
[2(x₁)]	=	1
[1^#(x₁)]	=	x₁ + 13507

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

1^#(4(0(1(2(x1)))))

→

1^#(4(0(3(1(x1)))))

(156)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

3^#(3(1(1(4(x1)))))	→	3^#(1(3(1(x1))))	(226)
3^#(0(4(5(1(x1)))))	→	5^#(0(0(3(4(1(x1))))))	(180)
0^#(1(5(3(1(x1)))))	→	3^#(1(5(4(1(x1)))))	(199)
0^#(1(5(3(1(x1)))))	→	0^#(3(1(5(4(1(x1))))))	(86)
3^#(0(1(0(2(x1)))))	→	0^#(4(1(x1)))	(124)
3^#(3(1(1(1(x1)))))	→	3^#(1(3(1(1(x1)))))	(185)
3^#(3(1(1(1(x1)))))	→	0^#(3(1(3(1(1(x1))))))	(210)
0^#(3(3(2(1(x1)))))	→	3^#(1(4(x1)))	(90)
5^#(3(0(1(1(x1)))))	→	3^#(1(1(5(0(x1)))))	(110)
5^#(3(0(1(1(x1)))))	→	0^#(3(1(1(5(0(x1))))))	(263)
3^#(3(0(1(4(x1)))))	→	3^#(1(4(x1)))	(79)
3^#(3(0(1(4(x1)))))	→	0^#(3(1(4(x1))))	(88)
3^#(3(0(1(4(x1)))))	→	0^#(0(3(1(4(x1)))))	(207)
3^#(3(0(1(4(x1)))))	→	3^#(0(0(3(1(4(x1))))))	(176)
0^#(5(4(3(1(x1)))))	→	3^#(4(1(4(5(x1)))))	(89)
0^#(5(4(3(1(x1)))))	→	0^#(3(4(1(4(5(x1))))))	(84)
5^#(4(3(4(1(x1)))))	→	3^#(4(4(1(5(x1)))))	(111)
5^#(4(3(4(1(x1)))))	→	0^#(3(4(4(1(5(x1))))))	(155)
0^#(1(0(1(2(x1)))))	→	0^#(2(1(1(3(x1)))))	(149)
0^#(1(0(1(2(x1)))))	→	0^#(0(2(1(1(3(x1))))))	(171)
3^#(0(1(1(x1))))	→	3^#(4(1(1(0(x1)))))	(112)
3^#(0(1(1(x1))))	→	0^#(3(4(1(1(0(x1))))))	(272)
3^#(5(0(2(1(x1)))))	→	5^#(1(x1))	(161)
3^#(5(0(2(1(x1)))))	→	0^#(5(1(x1)))	(80)
5^#(0(1(1(4(x1)))))	→	3^#(1(5(1(4(x1)))))	(128)
5^#(0(1(1(4(x1)))))	→	0^#(3(1(5(1(4(x1))))))	(131)
3^#(0(2(5(1(x1)))))	→	5^#(0(1(x1)))	(117)
0^#(2(3(1(x1))))	→	0^#(1(x1))	(243)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	1
[1(x₁)]	=	1
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	19505
[3(x₁)]	=	x₁ + 0
[4^#(x₁)]	=	0
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	19504
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	13507

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

3^#(0(4(5(1(x1)))))	→	5^#(0(0(3(4(1(x1))))))	(180)
5^#(3(0(1(1(x1)))))	→	3^#(1(1(5(0(x1)))))	(110)
5^#(3(0(1(1(x1)))))	→	0^#(3(1(1(5(0(x1))))))	(263)
5^#(4(3(4(1(x1)))))	→	3^#(4(4(1(5(x1)))))	(111)
5^#(4(3(4(1(x1)))))	→	0^#(3(4(4(1(5(x1))))))	(155)
3^#(5(0(2(1(x1)))))	→	5^#(1(x1))	(161)
3^#(5(0(2(1(x1)))))	→	0^#(5(1(x1)))	(80)
5^#(0(1(1(4(x1)))))	→	3^#(1(5(1(4(x1)))))	(128)
5^#(0(1(1(4(x1)))))	→	0^#(3(1(5(1(4(x1))))))	(131)
3^#(0(2(5(1(x1)))))	→	5^#(0(1(x1)))	(117)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

3^#(3(1(1(4(x1)))))	→	3^#(1(3(1(x1))))	(226)
0^#(1(5(3(1(x1)))))	→	3^#(1(5(4(1(x1)))))	(199)
0^#(1(5(3(1(x1)))))	→	0^#(3(1(5(4(1(x1))))))	(86)
3^#(0(1(0(2(x1)))))	→	0^#(4(1(x1)))	(124)
3^#(3(1(1(1(x1)))))	→	3^#(1(3(1(1(x1)))))	(185)
3^#(3(1(1(1(x1)))))	→	0^#(3(1(3(1(1(x1))))))	(210)
0^#(3(3(2(1(x1)))))	→	3^#(1(4(x1)))	(90)
3^#(3(0(1(4(x1)))))	→	3^#(1(4(x1)))	(79)
3^#(3(0(1(4(x1)))))	→	0^#(3(1(4(x1))))	(88)
3^#(3(0(1(4(x1)))))	→	0^#(0(3(1(4(x1)))))	(207)
3^#(3(0(1(4(x1)))))	→	3^#(0(0(3(1(4(x1))))))	(176)
0^#(5(4(3(1(x1)))))	→	3^#(4(1(4(5(x1)))))	(89)
0^#(5(4(3(1(x1)))))	→	0^#(3(4(1(4(5(x1))))))	(84)
0^#(1(0(1(2(x1)))))	→	0^#(2(1(1(3(x1)))))	(149)
0^#(1(0(1(2(x1)))))	→	0^#(0(2(1(1(3(x1))))))	(171)
3^#(0(1(1(x1))))	→	3^#(4(1(1(0(x1)))))	(112)
3^#(0(1(1(x1))))	→	0^#(3(4(1(1(0(x1))))))	(272)
0^#(2(3(1(x1))))	→	0^#(1(x1))	(243)

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	56581
[1(x₁)]	=	56580
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	56582
[3(x₁)]	=	56582
[4^#(x₁)]	=	0
[0(x₁)]	=	56582
[3^#(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	19504
[2(x₁)]	=	8702
[1^#(x₁)]	=	13507

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

3^#(3(1(1(4(x1)))))	→	3^#(1(3(1(x1))))	(226)
0^#(1(5(3(1(x1)))))	→	3^#(1(5(4(1(x1)))))	(199)
3^#(0(1(0(2(x1)))))	→	0^#(4(1(x1)))	(124)
3^#(3(1(1(1(x1)))))	→	3^#(1(3(1(1(x1)))))	(185)
3^#(3(1(1(1(x1)))))	→	0^#(3(1(3(1(1(x1))))))	(210)
0^#(3(3(2(1(x1)))))	→	3^#(1(4(x1)))	(90)
3^#(3(0(1(4(x1)))))	→	3^#(1(4(x1)))	(79)
3^#(3(0(1(4(x1)))))	→	0^#(3(1(4(x1))))	(88)
3^#(3(0(1(4(x1)))))	→	0^#(0(3(1(4(x1)))))	(207)
0^#(5(4(3(1(x1)))))	→	3^#(4(1(4(5(x1)))))	(89)
3^#(0(1(1(x1))))	→	3^#(4(1(1(0(x1)))))	(112)
3^#(0(1(1(x1))))	→	0^#(3(4(1(1(0(x1))))))	(272)

could be deleted.

1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

3^#(3(0(1(4(x1)))))

→

3^#(0(0(3(1(4(x1))))))

(176)

1.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[0^#(x₁)]

[1(x₁)]

[4(x₁)]

0	1
0	0

· x₁ +

[5(x₁)]

1	1
0	0

· x₁ +

[3(x₁)]

1	1
0	0

· x₁ +

[4^#(x₁)]

[0(x₁)]

1	0
0	0

· x₁ +

[3^#(x₁)]

1	0
1	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

0	1
0	0

· x₁ +

[1^#(x₁)]

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

3^#(3(0(1(4(x1)))))

→

3^#(0(0(3(1(4(x1))))))

(176)

could be deleted.

1.1.1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

0^#(1(5(3(1(x1)))))	→	0^#(3(1(5(4(1(x1))))))	(86)
0^#(5(4(3(1(x1)))))	→	0^#(3(4(1(4(5(x1))))))	(84)
0^#(1(0(1(2(x1)))))	→	0^#(2(1(1(3(x1)))))	(149)
0^#(1(0(1(2(x1)))))	→	0^#(0(2(1(1(3(x1))))))	(171)
0^#(2(3(1(x1))))	→	0^#(1(x1))	(243)

1.1.1.1.2.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 56581
[1(x₁)]	=	1
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	2
[3(x₁)]	=	x₁ + 0
[4^#(x₁)]	=	0
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	0
[5^#(x₁)]	=	19504
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	13507

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

0^#(5(4(3(1(x1)))))

→

0^#(3(4(1(4(5(x1))))))

(84)

could be deleted.

1.1.1.1.2.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(5(3(1(x1)))))	→	0^#(3(1(5(4(1(x1))))))	(86)
0^#(1(0(1(2(x1)))))	→	0^#(2(1(1(3(x1)))))	(149)
0^#(1(0(1(2(x1)))))	→	0^#(0(2(1(1(3(x1))))))	(171)
0^#(2(3(1(x1))))	→	0^#(1(x1))	(243)

1.1.1.1.2.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[0^#(x₁)]

0	1
0	0

· x₁ +

[1(x₁)]

8573

[4(x₁)]

8573

[5(x₁)]

0	0
1	0

· x₁ +

8573

[3(x₁)]

0	0
1	0

· x₁ +

8573

[4^#(x₁)]

[0(x₁)]

8573

[3^#(x₁)]

[5^#(x₁)]

[2(x₁)]

8573

[1^#(x₁)]

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

0^#(1(5(3(1(x1)))))

→

0^#(3(1(5(4(1(x1))))))

(86)

could be deleted.

1.1.1.1.2.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(0(1(2(x1)))))	→	0^#(2(1(1(3(x1)))))	(149)
0^#(1(0(1(2(x1)))))	→	0^#(0(2(1(1(3(x1))))))	(171)
0^#(2(3(1(x1))))	→	0^#(1(x1))	(243)

1.1.1.1.2.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[0^#(x₁)]

0	1
0	0

· x₁ +

[1(x₁)]

[4(x₁)]

0	1
0	0

· x₁ +

[5(x₁)]

1	0
0	0

· x₁ +

[3(x₁)]

0	0
1	0

· x₁ +

27810

[4^#(x₁)]

[0(x₁)]

0	1
0	1

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

[2(x₁)]

0	0
1	0

· x₁ +

[1^#(x₁)]

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

0^#(2(3(1(x1))))

→

0^#(1(x1))

(243)

could be deleted.

1.1.1.1.2.1.1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(0(1(2(x1)))))

→

0^#(0(2(1(1(3(x1))))))

(171)

1.1.1.1.2.1.1.1.2.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(0^#)	=	1
π(4)	=	1
π(0)	=	1
π(1^#)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(1)	=	2	status(1)	=	[]	list-extension(1)	=	Lex
prec(5)	=	2	status(5)	=	[]	list-extension(5)	=	Lex
prec(3)	=	2	status(3)	=	[]	list-extension(3)	=	Lex
prec(4^#)	=	0	status(4^#)	=	[]	list-extension(4^#)	=	Lex
prec(3^#)	=	0	status(3^#)	=	[]	list-extension(3^#)	=	Lex
prec(5^#)	=	0	status(5^#)	=	[]	list-extension(5^#)	=	Lex
prec(2)	=	0	status(2)	=	[]	list-extension(2)	=	Lex

and the following Max-polynomial interpretation

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

together with the usable rules

0(2(3(2(1(x1)))))	→	0(3(2(4(1(2(x1))))))	(18)
0(2(3(1(x1))))	→	4(0(3(2(1(x1)))))	(4)
0(1(2(3(1(x1)))))	→	1(4(0(3(1(2(x1))))))	(15)
3(0(1(2(x1))))	→	0(3(4(4(1(2(x1))))))	(8)
0(1(1(2(x1))))	→	0(1(0(3(1(2(x1))))))	(1)
0(2(3(1(x1))))	→	1(0(3(4(2(x1)))))	(3)
0(1(2(3(1(x1)))))	→	2(1(0(3(2(1(x1))))))	(16)
0(3(3(1(2(x1)))))	→	3(0(1(0(3(2(x1))))))	(21)
3(3(1(1(4(x1)))))	→	4(4(3(1(3(1(x1))))))	(36)
1(4(0(1(2(x1)))))	→	2(1(4(0(3(1(x1))))))	(26)
0(2(4(3(1(x1)))))	→	0(3(4(1(2(0(x1))))))	(19)
3(0(4(5(1(x1)))))	→	5(0(0(3(4(1(x1))))))	(32)
0(1(5(3(1(x1)))))	→	0(3(1(5(4(1(x1))))))	(17)
3(0(1(0(2(x1)))))	→	0(3(2(0(4(1(x1))))))	(27)
3(3(1(1(1(x1)))))	→	0(3(1(3(1(1(x1))))))	(34)
0(3(3(2(1(x1)))))	→	0(3(2(3(1(4(x1))))))	(22)
3(0(2(3(1(x1)))))	→	3(0(0(3(1(2(x1))))))	(28)
5(3(0(1(1(x1)))))	→	0(3(1(1(5(0(x1))))))	(44)
0(2(3(1(x1))))	→	0(0(3(2(1(4(x1))))))	(5)
3(3(0(1(4(x1)))))	→	3(0(0(3(1(4(x1))))))	(33)
3(0(2(1(x1))))	→	0(3(2(1(4(x1)))))	(10)
3(5(0(2(1(x1)))))	→	0(3(5(2(4(1(x1))))))	(39)
3(0(1(2(x1))))	→	0(3(4(1(2(4(x1))))))	(7)
0(2(4(3(1(x1)))))	→	1(0(0(3(4(2(x1))))))	(20)
0(5(4(3(1(x1)))))	→	0(3(4(1(4(5(x1))))))	(25)
5(4(3(4(1(x1)))))	→	0(3(4(4(1(5(x1))))))	(49)
3(0(2(5(1(x1)))))	→	0(3(5(2(1(4(x1))))))	(30)
0(1(2(3(1(x1)))))	→	1(0(3(1(2(2(x1))))))	(14)
3(0(2(5(1(x1)))))	→	5(4(0(3(2(1(x1))))))	(31)
0(1(0(1(2(x1)))))	→	0(0(2(1(1(3(x1))))))	(12)
5(3(0(2(1(x1)))))	→	0(3(4(2(1(5(x1))))))	(45)
0(5(2(3(1(x1)))))	→	0(3(2(1(5(4(x1))))))	(23)
0(5(2(3(1(x1)))))	→	0(5(0(3(2(1(x1))))))	(24)
3(0(2(1(x1))))	→	0(3(4(2(1(0(x1))))))	(11)
3(0(2(1(x1))))	→	0(3(2(1(0(x1)))))	(9)
0(1(1(4(2(x1)))))	→	1(0(3(1(4(2(x1))))))	(13)
4(0(1(1(4(x1)))))	→	1(0(3(4(4(1(x1))))))	(40)
3(0(1(1(x1))))	→	0(3(4(1(1(0(x1))))))	(6)
3(5(0(2(1(x1)))))	→	0(3(2(0(5(1(x1))))))	(38)
5(3(1(1(4(x1)))))	→	3(4(1(5(2(1(x1))))))	(48)
5(3(1(1(2(x1)))))	→	3(1(2(1(4(5(x1))))))	(47)
3(4(3(2(1(x1)))))	→	3(0(3(2(4(1(x1))))))	(37)
4(5(3(2(1(x1)))))	→	0(3(1(5(2(4(x1))))))	(41)
5(0(1(1(4(x1)))))	→	0(3(1(5(1(4(x1))))))	(42)
5(3(0(2(1(x1)))))	→	0(3(5(1(2(4(x1))))))	(46)
3(3(1(1(2(x1)))))	→	1(3(2(1(4(3(x1))))))	(35)
3(0(2(5(1(x1)))))	→	0(3(2(5(0(1(x1))))))	(29)
5(0(2(3(1(x1)))))	→	5(0(3(2(1(0(x1))))))	(43)
0(2(3(1(x1))))	→	0(3(2(0(1(x1)))))	(2)

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

0^#(1(0(1(2(x1)))))

→

0^#(0(2(1(1(3(x1))))))

(171)

could be deleted.

1.1.1.1.2.1.1.1.2.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.