Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213560)

The rewrite relation of the following TRS is considered.

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

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 162 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 111 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 + 0
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	x₁ + 0
[4^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 1
[3^#(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	x₁ + 0
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

0^#(0(4(1(0(x1)))))	→	3^#(0(0(x1)))	(208)
0^#(2(3(1(5(x1)))))	→	3^#(5(5(2(x1))))	(135)
3^#(1(5(0(2(x1)))))	→	5^#(x1)	(134)
0^#(1(5(3(5(x1)))))	→	3^#(5(3(5(x1))))	(132)
4^#(5(5(5(0(x1)))))	→	4^#(x1)	(205)
0^#(1(0(5(x1))))	→	3^#(5(0(x1)))	(203)
0^#(1(5(0(4(x1)))))	→	5^#(0(x1))	(123)
0^#(1(5(0(x1))))	→	3^#(5(0(x1)))	(197)
0^#(2(5(5(0(x1)))))	→	5^#(1(5(x1)))	(198)
5^#(3(1(0(x1))))	→	5^#(x1)	(112)
0^#(0(4(1(0(x1)))))	→	0^#(0(x1))	(113)
0^#(2(0(3(4(x1)))))	→	0^#(x1)	(192)
0^#(2(0(3(4(x1)))))	→	3^#(0(x1))	(105)
0^#(2(3(1(5(x1)))))	→	5^#(5(2(x1)))	(186)
0^#(1(5(0(4(x1)))))	→	0^#(x1)	(104)
0^#(3(1(0(x1))))	→	5^#(0(x1))	(102)
0^#(2(5(5(0(x1)))))	→	5^#(x1)	(100)
0^#(1(0(5(x1))))	→	5^#(0(x1))	(96)
4^#(0(1(0(x1))))	→	4^#(0(x1))	(176)
0^#(2(3(1(5(x1)))))	→	5^#(2(x1))	(91)
0^#(1(5(0(4(x1)))))	→	3^#(5(0(x1)))	(92)
0^#(0(1(2(x1))))	→	3^#(0(x1))	(170)
0^#(0(1(2(x1))))	→	0^#(x1)	(84)
3^#(1(3(2(0(x1)))))	→	5^#(2(x1))	(80)
0^#(2(0(3(4(x1)))))	→	4^#(5(2(3(0(x1)))))	(77)
0^#(2(0(3(4(x1)))))	→	5^#(2(3(0(x1))))	(75)
0^#(1(0(5(x1))))	→	0^#(x1)	(71)
5^#(3(1(0(x1))))	→	5^#(5(x1))	(160)
0^#(3(1(5(0(x1)))))	→	5^#(3(0(x1)))	(70)
0^#(5(3(2(0(x1)))))	→	3^#(5(x1))	(156)
3^#(1(2(0(x1))))	→	4^#(x1)	(66)
0^#(3(1(0(x1))))	→	3^#(5(0(x1)))	(65)
0^#(3(1(5(0(x1)))))	→	3^#(0(x1))	(151)
0^#(0(1(2(x1))))	→	3^#(3(0(x1)))	(59)
3^#(1(4(2(0(x1)))))	→	4^#(x1)	(144)
0^#(5(3(2(0(x1)))))	→	5^#(x1)	(55)
3^#(4(0(2(0(x1)))))	→	4^#(x1)	(140)
0^#(2(5(5(0(x1)))))	→	0^#(2(5(1(5(x1)))))	(141)
5^#(1(5(2(0(x1)))))	→	5^#(x1)	(52)
5^#(3(1(0(x1))))	→	3^#(5(5(x1)))	(139)
0^#(1(0(4(x1))))	→	3^#(0(4(x1)))	(49)

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

5^#(2(5(0(x1))))	→	3^#(5(0(x1)))	(152)
5^#(2(5(0(x1))))	→	5^#(2(3(5(0(x1)))))	(169)
3^#(4(2(0(x1))))	→	4^#(0(x1))	(81)
3^#(1(0(5(x1))))	→	0^#(x1)	(87)
3^#(1(0(5(x1))))	→	3^#(0(x1))	(85)
3^#(1(0(5(x1))))	→	5^#(3(0(x1)))	(143)
3^#(1(0(5(x1))))	→	3^#(5(3(0(x1))))	(199)
4^#(0(1(0(x1))))	→	0^#(4(0(x1)))	(168)
4^#(0(1(0(x1))))	→	3^#(0(4(0(x1))))	(200)
3^#(1(4(2(0(x1)))))	→	0^#(4(x1))	(179)
3^#(1(4(2(0(x1)))))	→	3^#(0(4(x1)))	(109)
3^#(1(4(2(0(x1)))))	→	3^#(3(0(4(x1))))	(125)
3^#(1(0(5(2(x1)))))	→	0^#(x1)	(195)
3^#(1(0(5(2(x1)))))	→	5^#(0(x1))	(74)
3^#(1(0(5(2(x1)))))	→	3^#(5(0(x1)))	(184)
5^#(1(0(5(x1))))	→	0^#(x1)	(153)
5^#(1(0(5(x1))))	→	5^#(0(x1))	(147)
5^#(1(0(5(x1))))	→	5^#(5(0(x1)))	(114)
5^#(1(0(5(x1))))	→	3^#(5(5(0(x1))))	(172)
5^#(1(0(5(x1))))	→	3^#(3(5(5(0(x1)))))	(86)
3^#(1(3(2(0(x1)))))	→	0^#(5(2(x1)))	(76)
3^#(1(3(2(0(x1)))))	→	3^#(0(5(2(x1))))	(127)
3^#(1(3(2(0(x1)))))	→	3^#(1(3(0(5(2(x1))))))	(90)
4^#(5(5(5(0(x1)))))	→	0^#(4(x1))	(204)
4^#(5(5(5(0(x1)))))	→	5^#(0(4(x1)))	(119)
4^#(5(5(5(0(x1)))))	→	5^#(1(5(0(4(x1)))))	(182)
4^#(5(5(5(0(x1)))))	→	5^#(5(1(5(0(4(x1))))))	(64)
3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
3^#(1(0(2(x1))))	→	0^#(x1)	(162)
3^#(1(0(2(x1))))	→	3^#(0(x1))	(54)
0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)
5^#(1(0(1(2(x1)))))	→	0^#(x1)	(171)
5^#(1(0(1(2(x1)))))	→	5^#(0(x1))	(89)
5^#(1(0(1(2(x1)))))	→	3^#(5(0(x1)))	(78)
3^#(1(2(0(x1))))	→	3^#(0(x1))	(68)
3^#(1(2(0(x1))))	→	3^#(3(0(x1)))	(56)
3^#(1(0(2(x1))))	→	5^#(0(2(x1)))	(142)
3^#(1(0(2(x1))))	→	3^#(5(0(2(x1))))	(83)
3^#(3(1(0(0(x1)))))	→	3^#(0(0(x1)))	(62)
3^#(3(1(0(0(x1)))))	→	3^#(3(0(0(x1))))	(187)
3^#(3(1(0(0(x1)))))	→	5^#(1(3(3(0(0(x1))))))	(53)
5^#(4(1(0(5(x1)))))	→	0^#(x1)	(161)
5^#(4(1(0(5(x1)))))	→	5^#(0(x1))	(150)
5^#(4(1(0(5(x1)))))	→	5^#(5(0(x1)))	(174)
5^#(4(1(0(5(x1)))))	→	3^#(5(5(0(x1))))	(121)
5^#(1(5(2(0(x1)))))	→	0^#(5(x1))	(158)
5^#(1(5(2(0(x1)))))	→	5^#(0(5(x1)))	(165)
5^#(1(5(2(0(x1)))))	→	3^#(5(0(5(x1))))	(181)
3^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(108)
3^#(1(5(0(2(x1)))))	→	3^#(0(5(x1)))	(111)
3^#(1(5(0(2(x1)))))	→	5^#(1(3(0(5(x1)))))	(159)
4^#(3(1(0(2(x1)))))	→	3^#(0(1(x1)))	(138)
5^#(1(1(0(4(x1)))))	→	0^#(x1)	(202)
5^#(1(1(0(4(x1)))))	→	3^#(0(x1))	(128)
5^#(1(1(0(4(x1)))))	→	4^#(3(0(x1)))	(99)
5^#(1(1(0(4(x1)))))	→	5^#(1(1(4(3(0(x1))))))	(120)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	42173
[1(x₁)]	=	1
[4(x₁)]	=	x₁ + 9577
[5(x₁)]	=	x₁ + 26157
[3(x₁)]	=	42173
[4^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	42173
[3^#(x₁)]	=	42173
[5^#(x₁)]	=	42173
[2(x₁)]	=	1

together with the usable rules

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

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

4^#(5(5(5(0(x1)))))	→	0^#(4(x1))	(204)
4^#(5(5(5(0(x1)))))	→	5^#(0(4(x1)))	(119)
4^#(5(5(5(0(x1)))))	→	5^#(1(5(0(4(x1)))))	(182)
4^#(5(5(5(0(x1)))))	→	5^#(5(1(5(0(4(x1))))))	(64)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(2(5(0(x1))))	→	3^#(5(0(x1)))	(152)
5^#(2(5(0(x1))))	→	5^#(2(3(5(0(x1)))))	(169)
3^#(4(2(0(x1))))	→	4^#(0(x1))	(81)
3^#(1(0(5(x1))))	→	0^#(x1)	(87)
3^#(1(0(5(x1))))	→	3^#(0(x1))	(85)
3^#(1(0(5(x1))))	→	5^#(3(0(x1)))	(143)
3^#(1(0(5(x1))))	→	3^#(5(3(0(x1))))	(199)
4^#(0(1(0(x1))))	→	0^#(4(0(x1)))	(168)
4^#(0(1(0(x1))))	→	3^#(0(4(0(x1))))	(200)
3^#(1(4(2(0(x1)))))	→	0^#(4(x1))	(179)
3^#(1(4(2(0(x1)))))	→	3^#(0(4(x1)))	(109)
3^#(1(4(2(0(x1)))))	→	3^#(3(0(4(x1))))	(125)
3^#(1(0(5(2(x1)))))	→	0^#(x1)	(195)
3^#(1(0(5(2(x1)))))	→	5^#(0(x1))	(74)
3^#(1(0(5(2(x1)))))	→	3^#(5(0(x1)))	(184)
5^#(1(0(5(x1))))	→	0^#(x1)	(153)
5^#(1(0(5(x1))))	→	5^#(0(x1))	(147)
5^#(1(0(5(x1))))	→	5^#(5(0(x1)))	(114)
5^#(1(0(5(x1))))	→	3^#(5(5(0(x1))))	(172)
5^#(1(0(5(x1))))	→	3^#(3(5(5(0(x1)))))	(86)
3^#(1(3(2(0(x1)))))	→	0^#(5(2(x1)))	(76)
3^#(1(3(2(0(x1)))))	→	3^#(0(5(2(x1))))	(127)
3^#(1(3(2(0(x1)))))	→	3^#(1(3(0(5(2(x1))))))	(90)
3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
3^#(1(0(2(x1))))	→	0^#(x1)	(162)
3^#(1(0(2(x1))))	→	3^#(0(x1))	(54)
0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)
5^#(1(0(1(2(x1)))))	→	0^#(x1)	(171)
5^#(1(0(1(2(x1)))))	→	5^#(0(x1))	(89)
5^#(1(0(1(2(x1)))))	→	3^#(5(0(x1)))	(78)
3^#(1(2(0(x1))))	→	3^#(0(x1))	(68)
3^#(1(2(0(x1))))	→	3^#(3(0(x1)))	(56)
3^#(1(0(2(x1))))	→	5^#(0(2(x1)))	(142)
3^#(1(0(2(x1))))	→	3^#(5(0(2(x1))))	(83)
3^#(3(1(0(0(x1)))))	→	3^#(0(0(x1)))	(62)
3^#(3(1(0(0(x1)))))	→	3^#(3(0(0(x1))))	(187)
3^#(3(1(0(0(x1)))))	→	5^#(1(3(3(0(0(x1))))))	(53)
5^#(4(1(0(5(x1)))))	→	0^#(x1)	(161)
5^#(4(1(0(5(x1)))))	→	5^#(0(x1))	(150)
5^#(4(1(0(5(x1)))))	→	5^#(5(0(x1)))	(174)
5^#(4(1(0(5(x1)))))	→	3^#(5(5(0(x1))))	(121)
5^#(1(5(2(0(x1)))))	→	0^#(5(x1))	(158)
5^#(1(5(2(0(x1)))))	→	5^#(0(5(x1)))	(165)
5^#(1(5(2(0(x1)))))	→	3^#(5(0(5(x1))))	(181)
3^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(108)
3^#(1(5(0(2(x1)))))	→	3^#(0(5(x1)))	(111)
3^#(1(5(0(2(x1)))))	→	5^#(1(3(0(5(x1)))))	(159)
4^#(3(1(0(2(x1)))))	→	3^#(0(1(x1)))	(138)
5^#(1(1(0(4(x1)))))	→	0^#(x1)	(202)
5^#(1(1(0(4(x1)))))	→	3^#(0(x1))	(128)
5^#(1(1(0(4(x1)))))	→	4^#(3(0(x1)))	(99)
5^#(1(1(0(4(x1)))))	→	5^#(1(1(4(3(0(x1))))))	(120)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	42173
[1(x₁)]	=	4301
[4(x₁)]	=	4302
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	4301
[4^#(x₁)]	=	42173
[0(x₁)]	=	4301
[3^#(x₁)]	=	42173
[5^#(x₁)]	=	x₁ + 37872
[2(x₁)]	=	4301

together with the usable rules

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

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

5^#(4(1(0(5(x1)))))	→	0^#(x1)	(161)
5^#(4(1(0(5(x1)))))	→	5^#(0(x1))	(150)
5^#(4(1(0(5(x1)))))	→	5^#(5(0(x1)))	(174)
5^#(4(1(0(5(x1)))))	→	3^#(5(5(0(x1))))	(121)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(2(5(0(x1))))	→	3^#(5(0(x1)))	(152)
5^#(2(5(0(x1))))	→	5^#(2(3(5(0(x1)))))	(169)
3^#(4(2(0(x1))))	→	4^#(0(x1))	(81)
3^#(1(0(5(x1))))	→	0^#(x1)	(87)
3^#(1(0(5(x1))))	→	3^#(0(x1))	(85)
3^#(1(0(5(x1))))	→	5^#(3(0(x1)))	(143)
3^#(1(0(5(x1))))	→	3^#(5(3(0(x1))))	(199)
4^#(0(1(0(x1))))	→	0^#(4(0(x1)))	(168)
4^#(0(1(0(x1))))	→	3^#(0(4(0(x1))))	(200)
3^#(1(4(2(0(x1)))))	→	0^#(4(x1))	(179)
3^#(1(4(2(0(x1)))))	→	3^#(0(4(x1)))	(109)
3^#(1(4(2(0(x1)))))	→	3^#(3(0(4(x1))))	(125)
3^#(1(0(5(2(x1)))))	→	0^#(x1)	(195)
3^#(1(0(5(2(x1)))))	→	5^#(0(x1))	(74)
3^#(1(0(5(2(x1)))))	→	3^#(5(0(x1)))	(184)
5^#(1(0(5(x1))))	→	0^#(x1)	(153)
5^#(1(0(5(x1))))	→	5^#(0(x1))	(147)
5^#(1(0(5(x1))))	→	5^#(5(0(x1)))	(114)
5^#(1(0(5(x1))))	→	3^#(5(5(0(x1))))	(172)
5^#(1(0(5(x1))))	→	3^#(3(5(5(0(x1)))))	(86)
3^#(1(3(2(0(x1)))))	→	0^#(5(2(x1)))	(76)
3^#(1(3(2(0(x1)))))	→	3^#(0(5(2(x1))))	(127)
3^#(1(3(2(0(x1)))))	→	3^#(1(3(0(5(2(x1))))))	(90)
3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
3^#(1(0(2(x1))))	→	0^#(x1)	(162)
3^#(1(0(2(x1))))	→	3^#(0(x1))	(54)
0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)
5^#(1(0(1(2(x1)))))	→	0^#(x1)	(171)
5^#(1(0(1(2(x1)))))	→	5^#(0(x1))	(89)
5^#(1(0(1(2(x1)))))	→	3^#(5(0(x1)))	(78)
3^#(1(2(0(x1))))	→	3^#(0(x1))	(68)
3^#(1(2(0(x1))))	→	3^#(3(0(x1)))	(56)
3^#(1(0(2(x1))))	→	5^#(0(2(x1)))	(142)
3^#(1(0(2(x1))))	→	3^#(5(0(2(x1))))	(83)
3^#(3(1(0(0(x1)))))	→	3^#(0(0(x1)))	(62)
3^#(3(1(0(0(x1)))))	→	3^#(3(0(0(x1))))	(187)
3^#(3(1(0(0(x1)))))	→	5^#(1(3(3(0(0(x1))))))	(53)
5^#(1(5(2(0(x1)))))	→	0^#(5(x1))	(158)
5^#(1(5(2(0(x1)))))	→	5^#(0(5(x1)))	(165)
5^#(1(5(2(0(x1)))))	→	3^#(5(0(5(x1))))	(181)
3^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(108)
3^#(1(5(0(2(x1)))))	→	3^#(0(5(x1)))	(111)
3^#(1(5(0(2(x1)))))	→	5^#(1(3(0(5(x1)))))	(159)
4^#(3(1(0(2(x1)))))	→	3^#(0(1(x1)))	(138)
5^#(1(1(0(4(x1)))))	→	0^#(x1)	(202)
5^#(1(1(0(4(x1)))))	→	3^#(0(x1))	(128)
5^#(1(1(0(4(x1)))))	→	4^#(3(0(x1)))	(99)
5^#(1(1(0(4(x1)))))	→	5^#(1(1(4(3(0(x1))))))	(120)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	60299
[1(x₁)]	=	18126
[4(x₁)]	=	18127
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	18126
[4^#(x₁)]	=	60299
[0(x₁)]	=	18126
[3^#(x₁)]	=	x₁ + 42173
[5^#(x₁)]	=	x₁ + 42173
[2(x₁)]	=	18126

together with the usable rules

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

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

3^#(4(2(0(x1))))

→

4^#(0(x1))

(81)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(2(5(0(x1))))	→	3^#(5(0(x1)))	(152)
5^#(2(5(0(x1))))	→	5^#(2(3(5(0(x1)))))	(169)
3^#(1(0(5(x1))))	→	0^#(x1)	(87)
3^#(1(0(5(x1))))	→	3^#(0(x1))	(85)
3^#(1(0(5(x1))))	→	5^#(3(0(x1)))	(143)
3^#(1(0(5(x1))))	→	3^#(5(3(0(x1))))	(199)
4^#(0(1(0(x1))))	→	0^#(4(0(x1)))	(168)
4^#(0(1(0(x1))))	→	3^#(0(4(0(x1))))	(200)
3^#(1(4(2(0(x1)))))	→	0^#(4(x1))	(179)
3^#(1(4(2(0(x1)))))	→	3^#(0(4(x1)))	(109)
3^#(1(4(2(0(x1)))))	→	3^#(3(0(4(x1))))	(125)
3^#(1(0(5(2(x1)))))	→	0^#(x1)	(195)
3^#(1(0(5(2(x1)))))	→	5^#(0(x1))	(74)
3^#(1(0(5(2(x1)))))	→	3^#(5(0(x1)))	(184)
5^#(1(0(5(x1))))	→	0^#(x1)	(153)
5^#(1(0(5(x1))))	→	5^#(0(x1))	(147)
5^#(1(0(5(x1))))	→	5^#(5(0(x1)))	(114)
5^#(1(0(5(x1))))	→	3^#(5(5(0(x1))))	(172)
5^#(1(0(5(x1))))	→	3^#(3(5(5(0(x1)))))	(86)
3^#(1(3(2(0(x1)))))	→	0^#(5(2(x1)))	(76)
3^#(1(3(2(0(x1)))))	→	3^#(0(5(2(x1))))	(127)
3^#(1(3(2(0(x1)))))	→	3^#(1(3(0(5(2(x1))))))	(90)
3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
3^#(1(0(2(x1))))	→	0^#(x1)	(162)
3^#(1(0(2(x1))))	→	3^#(0(x1))	(54)
0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)
5^#(1(0(1(2(x1)))))	→	0^#(x1)	(171)
5^#(1(0(1(2(x1)))))	→	5^#(0(x1))	(89)
5^#(1(0(1(2(x1)))))	→	3^#(5(0(x1)))	(78)
3^#(1(2(0(x1))))	→	3^#(0(x1))	(68)
3^#(1(2(0(x1))))	→	3^#(3(0(x1)))	(56)
3^#(1(0(2(x1))))	→	5^#(0(2(x1)))	(142)
3^#(1(0(2(x1))))	→	3^#(5(0(2(x1))))	(83)
3^#(3(1(0(0(x1)))))	→	3^#(0(0(x1)))	(62)
3^#(3(1(0(0(x1)))))	→	3^#(3(0(0(x1))))	(187)
3^#(3(1(0(0(x1)))))	→	5^#(1(3(3(0(0(x1))))))	(53)
5^#(1(5(2(0(x1)))))	→	0^#(5(x1))	(158)
5^#(1(5(2(0(x1)))))	→	5^#(0(5(x1)))	(165)
5^#(1(5(2(0(x1)))))	→	3^#(5(0(5(x1))))	(181)
3^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(108)
3^#(1(5(0(2(x1)))))	→	3^#(0(5(x1)))	(111)
3^#(1(5(0(2(x1)))))	→	5^#(1(3(0(5(x1)))))	(159)
4^#(3(1(0(2(x1)))))	→	3^#(0(1(x1)))	(138)
5^#(1(1(0(4(x1)))))	→	0^#(x1)	(202)
5^#(1(1(0(4(x1)))))	→	3^#(0(x1))	(128)
5^#(1(1(0(4(x1)))))	→	4^#(3(0(x1)))	(99)
5^#(1(1(0(4(x1)))))	→	5^#(1(1(4(3(0(x1))))))	(120)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	60300
[1(x₁)]	=	13957
[4(x₁)]	=	x₁ + 1
[5(x₁)]	=	x₁ + 1
[3(x₁)]	=	13958
[4^#(x₁)]	=	x₁ + 46342
[0(x₁)]	=	13959
[3^#(x₁)]	=	60300
[5^#(x₁)]	=	60300
[2(x₁)]	=	329

together with the usable rules

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

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

4^#(0(1(0(x1))))	→	0^#(4(0(x1)))	(168)
4^#(0(1(0(x1))))	→	3^#(0(4(0(x1))))	(200)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(2(5(0(x1))))	→	3^#(5(0(x1)))	(152)
5^#(2(5(0(x1))))	→	5^#(2(3(5(0(x1)))))	(169)
3^#(1(0(5(x1))))	→	0^#(x1)	(87)
3^#(1(0(5(x1))))	→	3^#(0(x1))	(85)
3^#(1(0(5(x1))))	→	5^#(3(0(x1)))	(143)
3^#(1(0(5(x1))))	→	3^#(5(3(0(x1))))	(199)
3^#(1(4(2(0(x1)))))	→	0^#(4(x1))	(179)
3^#(1(4(2(0(x1)))))	→	3^#(0(4(x1)))	(109)
3^#(1(4(2(0(x1)))))	→	3^#(3(0(4(x1))))	(125)
3^#(1(0(5(2(x1)))))	→	0^#(x1)	(195)
3^#(1(0(5(2(x1)))))	→	5^#(0(x1))	(74)
3^#(1(0(5(2(x1)))))	→	3^#(5(0(x1)))	(184)
5^#(1(0(5(x1))))	→	0^#(x1)	(153)
5^#(1(0(5(x1))))	→	5^#(0(x1))	(147)
5^#(1(0(5(x1))))	→	5^#(5(0(x1)))	(114)
5^#(1(0(5(x1))))	→	3^#(5(5(0(x1))))	(172)
5^#(1(0(5(x1))))	→	3^#(3(5(5(0(x1)))))	(86)
3^#(1(3(2(0(x1)))))	→	0^#(5(2(x1)))	(76)
3^#(1(3(2(0(x1)))))	→	3^#(0(5(2(x1))))	(127)
3^#(1(3(2(0(x1)))))	→	3^#(1(3(0(5(2(x1))))))	(90)
3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
3^#(1(0(2(x1))))	→	0^#(x1)	(162)
3^#(1(0(2(x1))))	→	3^#(0(x1))	(54)
0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)
5^#(1(0(1(2(x1)))))	→	0^#(x1)	(171)
5^#(1(0(1(2(x1)))))	→	5^#(0(x1))	(89)
5^#(1(0(1(2(x1)))))	→	3^#(5(0(x1)))	(78)
3^#(1(2(0(x1))))	→	3^#(0(x1))	(68)
3^#(1(2(0(x1))))	→	3^#(3(0(x1)))	(56)
3^#(1(0(2(x1))))	→	5^#(0(2(x1)))	(142)
3^#(1(0(2(x1))))	→	3^#(5(0(2(x1))))	(83)
3^#(3(1(0(0(x1)))))	→	3^#(0(0(x1)))	(62)
3^#(3(1(0(0(x1)))))	→	3^#(3(0(0(x1))))	(187)
3^#(3(1(0(0(x1)))))	→	5^#(1(3(3(0(0(x1))))))	(53)
5^#(1(5(2(0(x1)))))	→	0^#(5(x1))	(158)
5^#(1(5(2(0(x1)))))	→	5^#(0(5(x1)))	(165)
5^#(1(5(2(0(x1)))))	→	3^#(5(0(5(x1))))	(181)
3^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(108)
3^#(1(5(0(2(x1)))))	→	3^#(0(5(x1)))	(111)
3^#(1(5(0(2(x1)))))	→	5^#(1(3(0(5(x1)))))	(159)
4^#(3(1(0(2(x1)))))	→	3^#(0(1(x1)))	(138)
5^#(1(1(0(4(x1)))))	→	0^#(x1)	(202)
5^#(1(1(0(4(x1)))))	→	3^#(0(x1))	(128)
5^#(1(1(0(4(x1)))))	→	4^#(3(0(x1)))	(99)
5^#(1(1(0(4(x1)))))	→	5^#(1(1(4(3(0(x1))))))	(120)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	1
0	1

· x₁ +

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

[3(x₁)]

1	0
1	0

· x₁ +

[4^#(x₁)]

1	0
1	0

· x₁ +

[0(x₁)]

[3^#(x₁)]

1	0
1	0

· x₁ +

[5^#(x₁)]

x₁ +

[2(x₁)]

together with the usable rules

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

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

5^#(2(5(0(x1))))	→	3^#(5(0(x1)))	(152)
3^#(1(0(5(x1))))	→	0^#(x1)	(87)
3^#(1(0(5(x1))))	→	3^#(0(x1))	(85)
3^#(1(0(5(x1))))	→	5^#(3(0(x1)))	(143)
3^#(1(0(5(x1))))	→	3^#(5(3(0(x1))))	(199)
3^#(1(4(2(0(x1)))))	→	0^#(4(x1))	(179)
3^#(1(4(2(0(x1)))))	→	3^#(0(4(x1)))	(109)
3^#(1(4(2(0(x1)))))	→	3^#(3(0(4(x1))))	(125)
3^#(1(0(5(2(x1)))))	→	0^#(x1)	(195)
3^#(1(0(5(2(x1)))))	→	5^#(0(x1))	(74)
3^#(1(0(5(2(x1)))))	→	3^#(5(0(x1)))	(184)
5^#(1(0(5(x1))))	→	0^#(x1)	(153)
5^#(1(0(5(x1))))	→	5^#(0(x1))	(147)
5^#(1(0(5(x1))))	→	5^#(5(0(x1)))	(114)
5^#(1(0(5(x1))))	→	3^#(5(5(0(x1))))	(172)
5^#(1(0(5(x1))))	→	3^#(3(5(5(0(x1)))))	(86)
3^#(1(0(2(x1))))	→	0^#(x1)	(162)
3^#(1(0(2(x1))))	→	3^#(0(x1))	(54)
5^#(1(0(1(2(x1)))))	→	0^#(x1)	(171)
5^#(1(0(1(2(x1)))))	→	5^#(0(x1))	(89)
5^#(1(0(1(2(x1)))))	→	3^#(5(0(x1)))	(78)
3^#(1(0(2(x1))))	→	5^#(0(2(x1)))	(142)
3^#(1(0(2(x1))))	→	3^#(5(0(2(x1))))	(83)
3^#(3(1(0(0(x1)))))	→	3^#(0(0(x1)))	(62)
3^#(3(1(0(0(x1)))))	→	3^#(3(0(0(x1))))	(187)
3^#(3(1(0(0(x1)))))	→	5^#(1(3(3(0(0(x1))))))	(53)
5^#(1(5(2(0(x1)))))	→	0^#(5(x1))	(158)
5^#(1(5(2(0(x1)))))	→	5^#(0(5(x1)))	(165)
5^#(1(5(2(0(x1)))))	→	3^#(5(0(5(x1))))	(181)
3^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(108)
3^#(1(5(0(2(x1)))))	→	3^#(0(5(x1)))	(111)
3^#(1(5(0(2(x1)))))	→	5^#(1(3(0(5(x1)))))	(159)
4^#(3(1(0(2(x1)))))	→	3^#(0(1(x1)))	(138)
5^#(1(1(0(4(x1)))))	→	0^#(x1)	(202)
5^#(1(1(0(4(x1)))))	→	3^#(0(x1))	(128)
5^#(1(1(0(4(x1)))))	→	4^#(3(0(x1)))	(99)
5^#(1(1(0(4(x1)))))	→	5^#(1(1(4(3(0(x1))))))	(120)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

5^#(2(5(0(x1))))

→

5^#(2(3(5(0(x1)))))

(169)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	60301
[1(x₁)]	=	1
[4(x₁)]	=	x₁ + 1
[5(x₁)]	=	x₁ + 1
[3(x₁)]	=	2
[4^#(x₁)]	=	60300
[0(x₁)]	=	3
[3^#(x₁)]	=	60300
[5^#(x₁)]	=	x₁ + 60300
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

5^#(2(5(0(x1))))

→

5^#(2(3(5(0(x1)))))

(169)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

3^#(1(3(2(0(x1)))))	→	0^#(5(2(x1)))	(76)
3^#(1(3(2(0(x1)))))	→	3^#(0(5(2(x1))))	(127)
3^#(1(3(2(0(x1)))))	→	3^#(1(3(0(5(2(x1))))))	(90)
3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)
3^#(1(2(0(x1))))	→	3^#(0(x1))	(68)
3^#(1(2(0(x1))))	→	3^#(3(0(x1)))	(56)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	1
0	0

· x₁ +

[4(x₁)]

0	1
0	1

· x₁ +

[5(x₁)]

0	0
1	1

· x₁ +

[3(x₁)]

0	0
1	0

· x₁ +

[4^#(x₁)]

[0(x₁)]

10312

[3^#(x₁)]

1	0
0	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

0	1
0	1

· x₁ +

together with the usable rules

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

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

3^#(1(3(2(0(x1)))))	→	0^#(5(2(x1)))	(76)
3^#(1(3(2(0(x1)))))	→	3^#(0(5(2(x1))))	(127)
3^#(1(3(2(0(x1)))))	→	3^#(1(3(0(5(2(x1))))))	(90)
3^#(1(2(0(x1))))	→	3^#(0(x1))	(68)
3^#(1(2(0(x1))))	→	3^#(3(0(x1)))	(56)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 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)]

0	0
1	0

· x₁ +

[4(x₁)]

[5(x₁)]

1	0
0	0

· x₁ +

[3(x₁)]

0	1
0	0

· x₁ +

[4^#(x₁)]

[0(x₁)]

0	1
0	1

· x₁ +

[3^#(x₁)]

0	1
0	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

0	0
1	1

· x₁ +

together with the usable rules

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

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

0^#(2(0(3(4(x1)))))	→	0^#(4(5(2(3(0(x1))))))	(67)
0^#(2(5(5(0(x1)))))	→	0^#(0(2(5(1(5(x1))))))	(88)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

[1(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

0	0
1	0

· x₁ +

[5(x₁)]

1	0
0	0

· x₁ +

[3(x₁)]

0	1
0	0

· x₁ +

[4^#(x₁)]

[0(x₁)]

1	1
1	1

· x₁ +

19695

[3^#(x₁)]

0	1
0	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

0	1
1	0

· x₁ +

together with the usable rules

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

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

3^#(1(3(0(2(x1)))))	→	0^#(x1)	(60)
0^#(3(1(5(0(x1)))))	→	0^#(5(3(0(x1))))	(110)
0^#(3(1(5(0(x1)))))	→	3^#(0(5(3(0(x1)))))	(118)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

3^#(1(3(0(2(x1)))))	→	3^#(0(x1))	(107)
3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	1
1	1

· x₁ +

6714

[4(x₁)]

x₁ +

6715

[5(x₁)]

0	0
0	1

· x₁ +

[3(x₁)]

1	0
1	0

· x₁ +

[4^#(x₁)]

[0(x₁)]

6716

[3^#(x₁)]

0	1
0	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

together with the usable rules

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

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

3^#(1(3(0(2(x1)))))	→	3^#(3(0(x1)))	(148)
3^#(1(3(0(2(x1)))))	→	3^#(3(3(0(x1))))	(146)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

3^#(0(x1))

(107)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

14738

[5(x₁)]

x₁ +

[3(x₁)]

0	1
0	1

· x₁ +

[4^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

[3^#(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

[2(x₁)]

1	0
1	0

· x₁ +

together with the usable rules

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

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

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

→

3^#(0(x1))

(107)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

0^#(1(5(0(x1))))	→	0^#(1(3(5(0(x1)))))	(191)
0^#(1(0(5(x1))))	→	0^#(1(3(5(0(x1)))))	(167)
0^#(1(5(3(5(x1)))))	→	0^#(1(3(5(3(5(x1))))))	(98)
0^#(1(0(4(x1))))	→	0^#(1(3(0(4(x1)))))	(193)

1.1.1.1.2 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	1

· x₁ +

[1(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

[5(x₁)]

x₁ +

[3(x₁)]

0	1
0	1

· x₁ +

[4^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

[3^#(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

[2(x₁)]

1	0
1	0

· x₁ +

together with the usable rules

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

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

0^#(1(5(0(x1))))	→	0^#(1(3(5(0(x1)))))	(191)
0^#(1(0(5(x1))))	→	0^#(1(3(5(0(x1)))))	(167)
0^#(1(0(4(x1))))	→	0^#(1(3(0(4(x1)))))	(193)

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

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

→

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

(98)

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

[1(x₁)]

0	1
0	1

· x₁ +

[4(x₁)]

[5(x₁)]

[3(x₁)]

1	0
1	0

· x₁ +

[4^#(x₁)]

[0(x₁)]

0	0
1	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

[2(x₁)]

together with the usable rules

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

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

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

→

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

(98)

could be deleted.

1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.