Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212774)

The rewrite relation of the following TRS is considered.

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

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 171 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

0^#(5(4(2(x1))))	→	5^#(x1)	(142)
0^#(1(2(3(x1))))	→	0^#(2(0(1(3(x1)))))	(140)
0^#(2(1(2(2(x1)))))	→	2^#(0(2(2(1(x1)))))	(215)
2^#(5(0(0(3(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(138)
0^#(1(2(2(x1))))	→	0^#(x1)	(86)
0^#(5(0(1(2(x1)))))	→	0^#(0(2(0(1(5(x1))))))	(135)
5^#(2(0(1(2(x1)))))	→	2^#(2(x1))	(131)
0^#(2(1(2(2(x1)))))	→	2^#(2(1(x1)))	(130)
2^#(1(0(4(x1))))	→	2^#(2(2(x1)))	(129)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
0^#(2(1(2(2(x1)))))	→	0^#(2(0(2(2(1(x1))))))	(126)
0^#(5(0(1(2(x1)))))	→	0^#(2(0(1(5(x1)))))	(124)
0^#(5(0(0(3(x1)))))	→	0^#(5(x1))	(208)
0^#(4(1(2(x1))))	→	0^#(1(x1))	(121)
0^#(1(2(2(x1))))	→	2^#(2(0(x1)))	(206)
5^#(0(1(2(2(x1)))))	→	0^#(2(x1))	(205)
0^#(2(1(2(x1))))	→	0^#(1(x1))	(119)
0^#(3(2(2(x1))))	→	0^#(2(2(x1)))	(203)
0^#(5(0(5(x1))))	→	5^#(5(x1))	(118)
0^#(0(5(4(2(x1)))))	→	5^#(x1)	(117)
0^#(5(0(0(3(x1)))))	→	0^#(3(0(5(x1))))	(201)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(2(1(2(2(x1)))))	→	0^#(2(2(1(x1))))	(116)
0^#(1(2(2(x1))))	→	0^#(2(2(1(0(x1)))))	(200)
0^#(5(0(5(x1))))	→	0^#(5(5(x1)))	(199)
0^#(2(3(0(5(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(113)
2^#(1(0(4(x1))))	→	2^#(2(x1))	(112)
0^#(1(2(2(x1))))	→	2^#(1(0(x1)))	(197)
0^#(4(5(2(3(x1)))))	→	5^#(x1)	(196)
0^#(2(1(2(x1))))	→	2^#(2(1(x1)))	(110)
0^#(1(2(3(x1))))	→	2^#(0(1(3(x1))))	(108)
0^#(5(0(5(x1))))	→	2^#(0(5(5(x1))))	(107)
0^#(4(1(2(x1))))	→	2^#(0(1(x1)))	(192)
0^#(1(2(2(2(x1)))))	→	2^#(2(2(1(2(x1)))))	(190)
2^#(5(0(0(3(x1)))))	→	2^#(0(0(5(3(x1)))))	(188)
2^#(1(0(4(x1))))	→	2^#(x1)	(186)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(5(0(1(2(x1)))))	→	2^#(0(1(5(x1))))	(181)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
0^#(4(1(2(x1))))	→	0^#(2(0(1(x1))))	(95)
2^#(1(0(4(x1))))	→	0^#(2(2(2(x1))))	(93)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
5^#(2(0(1(2(x1)))))	→	0^#(2(2(x1)))	(92)
0^#(5(2(5(x1))))	→	5^#(5(x1))	(177)
5^#(0(1(2(2(x1)))))	→	0^#(2(0(2(x1))))	(89)
0^#(5(0(0(3(x1)))))	→	5^#(x1)	(175)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(1(2(3(2(x1)))))	→	2^#(2(x1))	(173)
0^#(1(2(2(x1))))	→	0^#(x1)	(86)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)
0^#(1(2(2(x1))))	→	0^#(2(2(0(x1))))	(85)
0^#(1(2(2(2(x1)))))	→	2^#(1(2(x1)))	(170)
0^#(5(0(0(3(x1)))))	→	0^#(2(0(3(0(5(x1))))))	(80)
0^#(1(2(2(2(x1)))))	→	0^#(2(2(2(1(2(x1))))))	(169)
0^#(5(0(1(2(x1)))))	→	5^#(x1)	(167)
0^#(5(0(5(x1))))	→	0^#(2(0(5(5(x1)))))	(75)
2^#(1(0(0(4(x1)))))	→	2^#(x1)	(73)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(0(0(3(x1)))))	→	2^#(0(3(0(5(x1)))))	(72)
0^#(5(4(2(x1))))	→	2^#(0(5(x1)))	(165)
0^#(2(3(0(5(x1)))))	→	2^#(0(0(5(3(x1)))))	(70)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
0^#(2(3(0(5(x1)))))	→	0^#(5(3(x1)))	(67)
2^#(1(0(0(4(x1)))))	→	0^#(2(x1))	(68)
0^#(2(1(2(x1))))	→	2^#(0(1(x1)))	(160)
2^#(5(4(2(x1))))	→	5^#(x1)	(66)
0^#(0(5(4(2(x1)))))	→	0^#(4(0(0(2(5(x1))))))	(62)
0^#(2(3(0(5(x1)))))	→	0^#(0(5(3(x1))))	(156)
0^#(5(0(1(2(x1)))))	→	0^#(1(5(x1)))	(155)
0^#(2(1(2(x1))))	→	2^#(2(2(1(x1))))	(153)
0^#(5(1(4(2(x1)))))	→	5^#(x1)	(60)
0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(2(1(2(x1))))	→	2^#(1(x1))	(59)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
0^#(2(1(2(x1))))	→	0^#(2(2(2(1(x1)))))	(151)
0^#(2(1(2(x1))))	→	0^#(2(2(0(1(x1)))))	(56)
0^#(1(2(3(2(x1)))))	→	0^#(2(2(x1)))	(149)
2^#(1(0(0(4(x1)))))	→	0^#(0(2(x1)))	(55)
0^#(1(2(2(x1))))	→	2^#(0(x1))	(148)
0^#(2(1(2(2(x1)))))	→	2^#(1(x1))	(53)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
5^#(0(1(2(2(x1)))))	→	2^#(0(2(x1)))	(146)
0^#(5(4(2(x1))))	→	2^#(2(0(5(x1))))	(144)
0^#(1(2(2(2(x1)))))	→	2^#(2(1(2(x1))))	(49)
0^#(2(1(2(x1))))	→	2^#(2(0(1(x1))))	(50)
0^#(1(2(2(x1))))	→	2^#(2(1(0(x1))))	(48)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 29535
[3(x₁)]	=	x₁ + 0
[2^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	x₁ + 1
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

0^#(5(4(2(x1))))	→	5^#(x1)	(142)
5^#(2(0(1(2(x1)))))	→	2^#(2(x1))	(131)
5^#(0(1(2(2(x1)))))	→	0^#(2(x1))	(205)
0^#(5(0(5(x1))))	→	5^#(5(x1))	(118)
0^#(0(5(4(2(x1)))))	→	5^#(x1)	(117)
0^#(4(5(2(3(x1)))))	→	5^#(x1)	(196)
5^#(2(0(1(2(x1)))))	→	0^#(2(2(x1)))	(92)
0^#(5(2(5(x1))))	→	5^#(5(x1))	(177)
5^#(0(1(2(2(x1)))))	→	0^#(2(0(2(x1))))	(89)
0^#(5(0(0(3(x1)))))	→	5^#(x1)	(175)
0^#(5(0(1(2(x1)))))	→	5^#(x1)	(167)
2^#(5(4(2(x1))))	→	5^#(x1)	(66)
0^#(5(1(4(2(x1)))))	→	5^#(x1)	(60)
5^#(0(1(2(2(x1)))))	→	2^#(0(2(x1)))	(146)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(2(3(x1))))	→	2^#(0(1(3(x1))))	(108)
0^#(1(2(3(x1))))	→	0^#(2(0(1(3(x1)))))	(140)
0^#(1(2(2(x1))))	→	0^#(x1)	(86)
0^#(1(2(2(x1))))	→	2^#(0(x1))	(148)
0^#(1(2(2(x1))))	→	2^#(2(0(x1)))	(206)
0^#(1(2(2(x1))))	→	0^#(2(2(0(x1))))	(85)
0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(0(0(3(x1)))))	→	0^#(5(x1))	(208)
0^#(5(0(0(3(x1)))))	→	0^#(3(0(5(x1))))	(201)
0^#(5(0(0(3(x1)))))	→	2^#(0(3(0(5(x1)))))	(72)
0^#(5(0(0(3(x1)))))	→	0^#(2(0(3(0(5(x1))))))	(80)
0^#(1(2(3(2(x1)))))	→	2^#(2(x1))	(173)
0^#(1(2(3(2(x1)))))	→	0^#(2(2(x1)))	(149)
2^#(1(0(4(x1))))	→	2^#(x1)	(186)
2^#(1(0(4(x1))))	→	2^#(2(x1))	(112)
2^#(1(0(4(x1))))	→	2^#(2(2(x1)))	(129)
2^#(1(0(4(x1))))	→	0^#(2(2(2(x1))))	(93)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(5(4(2(x1))))	→	2^#(0(5(x1)))	(165)
0^#(5(4(2(x1))))	→	2^#(2(0(5(x1))))	(144)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
0^#(0(5(4(2(x1)))))	→	0^#(4(0(0(2(5(x1))))))	(62)
0^#(2(1(2(x1))))	→	2^#(1(x1))	(59)
0^#(2(1(2(x1))))	→	2^#(2(1(x1)))	(110)
0^#(2(1(2(x1))))	→	2^#(2(2(1(x1))))	(153)
0^#(2(1(2(x1))))	→	0^#(2(2(2(1(x1)))))	(151)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
0^#(1(2(2(2(x1)))))	→	2^#(1(2(x1)))	(170)
0^#(1(2(2(2(x1)))))	→	2^#(2(1(2(x1))))	(49)
0^#(1(2(2(2(x1)))))	→	2^#(2(2(1(2(x1)))))	(190)
0^#(1(2(2(2(x1)))))	→	0^#(2(2(2(1(2(x1))))))	(169)
0^#(2(1(2(2(x1)))))	→	2^#(1(x1))	(53)
0^#(2(1(2(2(x1)))))	→	2^#(2(1(x1)))	(130)
0^#(2(1(2(2(x1)))))	→	0^#(2(2(1(x1))))	(116)
0^#(2(1(2(2(x1)))))	→	2^#(0(2(2(1(x1)))))	(215)
0^#(2(1(2(2(x1)))))	→	0^#(2(0(2(2(1(x1))))))	(126)
0^#(5(0(5(x1))))	→	0^#(5(5(x1)))	(199)
0^#(5(0(5(x1))))	→	2^#(0(5(5(x1))))	(107)
0^#(5(0(5(x1))))	→	0^#(2(0(5(5(x1)))))	(75)
0^#(2(3(0(5(x1)))))	→	0^#(5(3(x1)))	(67)
0^#(2(3(0(5(x1)))))	→	0^#(0(5(3(x1))))	(156)
0^#(2(3(0(5(x1)))))	→	2^#(0(0(5(3(x1)))))	(70)
0^#(2(3(0(5(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(113)
0^#(4(1(2(x1))))	→	0^#(1(x1))	(121)
0^#(4(1(2(x1))))	→	2^#(0(1(x1)))	(192)
0^#(4(1(2(x1))))	→	0^#(2(0(1(x1))))	(95)
0^#(3(2(2(x1))))	→	0^#(2(2(x1)))	(203)
2^#(1(0(0(4(x1)))))	→	2^#(x1)	(73)
2^#(1(0(0(4(x1)))))	→	0^#(2(x1))	(68)
2^#(1(0(0(4(x1)))))	→	0^#(0(2(x1)))	(55)
0^#(2(1(2(x1))))	→	0^#(1(x1))	(119)
0^#(2(1(2(x1))))	→	2^#(0(1(x1)))	(160)
0^#(2(1(2(x1))))	→	2^#(2(0(1(x1))))	(50)
0^#(2(1(2(x1))))	→	0^#(2(2(0(1(x1)))))	(56)
0^#(5(0(1(2(x1)))))	→	0^#(1(5(x1)))	(155)
0^#(5(0(1(2(x1)))))	→	2^#(0(1(5(x1))))	(181)
0^#(5(0(1(2(x1)))))	→	0^#(2(0(1(5(x1)))))	(124)
0^#(5(0(1(2(x1)))))	→	0^#(0(2(0(1(5(x1))))))	(135)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)
2^#(5(0(0(3(x1)))))	→	2^#(0(0(5(3(x1)))))	(188)
2^#(5(0(0(3(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(138)
0^#(1(2(2(x1))))	→	0^#(x1)	(86)
0^#(1(2(2(x1))))	→	2^#(1(0(x1)))	(197)
0^#(1(2(2(x1))))	→	2^#(2(1(0(x1))))	(48)
0^#(1(2(2(x1))))	→	0^#(2(2(1(0(x1)))))	(200)

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)]

0	1
0	0

· x₁ +

[1(x₁)]

1	0
1	0

· x₁ +

39944

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

61230

101175

[3(x₁)]

0	0
1	0

· x₁ +

[2^#(x₁)]

0	1
0	0

· x₁ +

[0(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

[2(x₁)]

1	1
0	1

· x₁ +

10693

together with the usable rules

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

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

0^#(1(2(3(x1))))	→	2^#(0(1(3(x1))))	(108)
0^#(1(2(3(x1))))	→	0^#(2(0(1(3(x1)))))	(140)
0^#(1(2(2(x1))))	→	0^#(x1)	(86)
0^#(1(2(2(x1))))	→	2^#(0(x1))	(148)
0^#(1(2(2(x1))))	→	2^#(2(0(x1)))	(206)
0^#(1(2(2(x1))))	→	0^#(2(2(0(x1))))	(85)
0^#(1(2(3(2(x1)))))	→	2^#(2(x1))	(173)
0^#(1(2(3(2(x1)))))	→	0^#(2(2(x1)))	(149)
2^#(1(0(4(x1))))	→	2^#(x1)	(186)
2^#(1(0(4(x1))))	→	2^#(2(x1))	(112)
2^#(1(0(4(x1))))	→	2^#(2(2(x1)))	(129)
2^#(1(0(4(x1))))	→	0^#(2(2(2(x1))))	(93)
0^#(2(1(2(x1))))	→	2^#(1(x1))	(59)
0^#(2(1(2(x1))))	→	2^#(2(1(x1)))	(110)
0^#(2(1(2(x1))))	→	2^#(2(2(1(x1))))	(153)
0^#(2(1(2(x1))))	→	0^#(2(2(2(1(x1)))))	(151)
0^#(1(2(2(2(x1)))))	→	2^#(1(2(x1)))	(170)
0^#(1(2(2(2(x1)))))	→	2^#(2(1(2(x1))))	(49)
0^#(1(2(2(2(x1)))))	→	2^#(2(2(1(2(x1)))))	(190)
0^#(1(2(2(2(x1)))))	→	0^#(2(2(2(1(2(x1))))))	(169)
0^#(2(1(2(2(x1)))))	→	2^#(1(x1))	(53)
0^#(2(1(2(2(x1)))))	→	2^#(2(1(x1)))	(130)
0^#(2(1(2(2(x1)))))	→	0^#(2(2(1(x1))))	(116)
0^#(2(1(2(2(x1)))))	→	2^#(0(2(2(1(x1)))))	(215)
0^#(2(1(2(2(x1)))))	→	0^#(2(0(2(2(1(x1))))))	(126)
0^#(4(1(2(x1))))	→	0^#(1(x1))	(121)
0^#(4(1(2(x1))))	→	2^#(0(1(x1)))	(192)
0^#(4(1(2(x1))))	→	0^#(2(0(1(x1))))	(95)
0^#(3(2(2(x1))))	→	0^#(2(2(x1)))	(203)
2^#(1(0(0(4(x1)))))	→	2^#(x1)	(73)
2^#(1(0(0(4(x1)))))	→	0^#(2(x1))	(68)
2^#(1(0(0(4(x1)))))	→	0^#(0(2(x1)))	(55)
0^#(2(1(2(x1))))	→	0^#(1(x1))	(119)
0^#(2(1(2(x1))))	→	2^#(0(1(x1)))	(160)
0^#(2(1(2(x1))))	→	2^#(2(0(1(x1))))	(50)
0^#(2(1(2(x1))))	→	0^#(2(2(0(1(x1)))))	(56)
0^#(5(0(1(2(x1)))))	→	0^#(1(5(x1)))	(155)
0^#(5(0(1(2(x1)))))	→	2^#(0(1(5(x1))))	(181)
0^#(5(0(1(2(x1)))))	→	0^#(2(0(1(5(x1)))))	(124)
0^#(5(0(1(2(x1)))))	→	0^#(0(2(0(1(5(x1))))))	(135)
0^#(1(2(2(x1))))	→	0^#(x1)	(86)
0^#(1(2(2(x1))))	→	2^#(1(0(x1)))	(197)
0^#(1(2(2(x1))))	→	2^#(2(1(0(x1))))	(48)
0^#(1(2(2(x1))))	→	0^#(2(2(1(0(x1)))))	(200)

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

0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(0(0(3(x1)))))	→	0^#(5(x1))	(208)
0^#(5(0(0(3(x1)))))	→	2^#(0(3(0(5(x1)))))	(72)
0^#(5(0(0(3(x1)))))	→	0^#(2(0(3(0(5(x1))))))	(80)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(5(4(2(x1))))	→	2^#(0(5(x1)))	(165)
0^#(5(4(2(x1))))	→	2^#(2(0(5(x1))))	(144)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
0^#(5(0(5(x1))))	→	0^#(5(5(x1)))	(199)
0^#(5(0(5(x1))))	→	2^#(0(5(5(x1))))	(107)
0^#(5(0(5(x1))))	→	0^#(2(0(5(5(x1)))))	(75)
0^#(2(3(0(5(x1)))))	→	0^#(5(3(x1)))	(67)
0^#(2(3(0(5(x1)))))	→	0^#(0(5(3(x1))))	(156)
0^#(2(3(0(5(x1)))))	→	2^#(0(0(5(3(x1)))))	(70)
0^#(2(3(0(5(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(113)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)
2^#(5(0(0(3(x1)))))	→	2^#(0(0(5(3(x1)))))	(188)
2^#(5(0(0(3(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(138)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

0^#(2(3(0(5(x1)))))	→	0^#(5(3(x1)))	(67)
0^#(2(3(0(5(x1)))))	→	0^#(0(5(3(x1))))	(156)
0^#(2(3(0(5(x1)))))	→	2^#(0(0(5(3(x1)))))	(70)
0^#(2(3(0(5(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(113)

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

0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(0(0(3(x1)))))	→	0^#(5(x1))	(208)
0^#(5(0(0(3(x1)))))	→	2^#(0(3(0(5(x1)))))	(72)
0^#(5(0(0(3(x1)))))	→	0^#(2(0(3(0(5(x1))))))	(80)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(5(4(2(x1))))	→	2^#(0(5(x1)))	(165)
0^#(5(4(2(x1))))	→	2^#(2(0(5(x1))))	(144)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
0^#(5(0(5(x1))))	→	0^#(5(5(x1)))	(199)
0^#(5(0(5(x1))))	→	2^#(0(5(5(x1))))	(107)
0^#(5(0(5(x1))))	→	0^#(2(0(5(5(x1)))))	(75)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)
2^#(5(0(0(3(x1)))))	→	2^#(0(0(5(3(x1)))))	(188)
2^#(5(0(0(3(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(138)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	100853
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	56023
[5(x₁)]	=	100853
[3(x₁)]	=	x₁ + 44829
[2^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	100852
[5^#(x₁)]	=	1
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

0^#(5(0(0(3(x1)))))	→	2^#(0(3(0(5(x1)))))	(72)
0^#(5(4(2(x1))))	→	2^#(0(5(x1)))	(165)
0^#(5(4(2(x1))))	→	2^#(2(0(5(x1))))	(144)
0^#(5(0(5(x1))))	→	2^#(0(5(5(x1))))	(107)
2^#(5(0(0(3(x1)))))	→	2^#(0(0(5(3(x1)))))	(188)

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

0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(0(0(3(x1)))))	→	0^#(5(x1))	(208)
0^#(5(0(0(3(x1)))))	→	0^#(2(0(3(0(5(x1))))))	(80)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
0^#(5(0(5(x1))))	→	0^#(5(5(x1)))	(199)
0^#(5(0(5(x1))))	→	0^#(2(0(5(5(x1)))))	(75)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)
2^#(5(0(0(3(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(138)

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	0
0	0

· x₁ +

[1(x₁)]

[4(x₁)]

[5(x₁)]

34648

12791

[3(x₁)]

[2^#(x₁)]

34648

[0(x₁)]

34648

[5^#(x₁)]

[2(x₁)]

0	1
0	0

· x₁ +

21857

12791

together with the usable rules

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

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

0^#(5(0(0(3(x1)))))	→	0^#(2(0(3(0(5(x1))))))	(80)
0^#(5(0(5(x1))))	→	0^#(2(0(5(5(x1)))))	(75)
2^#(5(0(0(3(x1)))))	→	0^#(2(0(0(5(3(x1))))))	(138)

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

0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(0(0(3(x1)))))	→	0^#(5(x1))	(208)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
0^#(5(0(5(x1))))	→	0^#(5(5(x1)))	(199)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)

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)]

0	1
0	1

· x₁ +

34648

[1(x₁)]

0	1
1	0

· x₁ +

[4(x₁)]

1	0
0	0

· x₁ +

[5(x₁)]

0	0
1	0

· x₁ +

[3(x₁)]

0	0
1	1

· x₁ +

[2^#(x₁)]

34648

[0(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

[2(x₁)]

1	1
0	0

· x₁ +

together with the usable rules

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

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

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

→

0^#(5(x1))

(208)

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 1 component.

The 1^st component contains the pair

0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
0^#(5(0(5(x1))))	→	0^#(5(5(x1)))	(199)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

34648

[1(x₁)]

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

1	1
1	1

· x₁ +

[3(x₁)]

[2^#(x₁)]

1	0
0	0

· x₁ +

43162

[0(x₁)]

1	0
1	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

1	0
1	1

· x₁ +

8515

together with the usable rules

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

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

0^#(5(2(5(x1))))	→	2^#(5(5(x1)))	(58)
0^#(5(2(5(x1))))	→	2^#(2(5(5(x1))))	(162)
0^#(5(2(5(x1))))	→	0^#(2(2(5(5(x1)))))	(71)
0^#(5(4(2(x1))))	→	0^#(5(x1))	(115)
0^#(5(4(2(x1))))	→	0^#(2(2(0(5(x1)))))	(184)
0^#(0(5(4(2(x1)))))	→	2^#(5(x1))	(179)
0^#(0(5(4(2(x1)))))	→	0^#(2(5(x1)))	(87)
0^#(0(5(4(2(x1)))))	→	0^#(0(2(5(x1))))	(57)
2^#(5(4(2(x1))))	→	2^#(5(x1))	(97)
2^#(5(4(2(x1))))	→	2^#(2(5(x1)))	(128)
2^#(5(4(2(x1))))	→	0^#(2(2(5(x1))))	(145)
2^#(5(0(0(3(x1)))))	→	0^#(5(3(x1)))	(163)
2^#(5(0(0(3(x1)))))	→	0^#(0(5(3(x1))))	(172)

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 1 component.

The 1^st component contains the pair

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

→

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

(199)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(2^#)

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

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

and the following Max-polynomial interpretation

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

together with the usable rules

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

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

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

→

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

(199)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.