Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/214169)

The rewrite relation of the following TRS is considered.

0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)

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 148 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

5^#(5(0(4(x1))))	→	0^#(2(5(4(2(x1)))))	(198)
0^#(1(2(0(x1))))	→	1^#(0(x1))	(196)
1^#(0(1(4(x1))))	→	1^#(4(0(x1)))	(197)
5^#(4(1(5(x1))))	→	5^#(2(1(x1)))	(130)
1^#(0(1(4(x1))))	→	0^#(x1)	(128)
1^#(0(1(4(x1))))	→	1^#(1(4(0(x1))))	(129)
4^#(1(0(4(x1))))	→	4^#(0(2(1(x1))))	(126)
4^#(0(0(3(1(x1)))))	→	0^#(x1)	(127)
0^#(3(4(0(x1))))	→	0^#(3(x1))	(192)
5^#(4(2(0(1(x1)))))	→	0^#(2(1(x1)))	(191)
1^#(2(4(3(1(x1)))))	→	1^#(4(x1))	(190)
0^#(0(1(x1)))	→	0^#(2(1(3(x1))))	(121)
0^#(1(2(0(x1))))	→	0^#(2(1(0(x1))))	(118)
0^#(5(4(1(1(x1)))))	→	4^#(0(2(5(1(1(x1))))))	(119)
5^#(0(5(1(x1))))	→	1^#(0(x1))	(116)
0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
4^#(1(0(0(x1))))	→	1^#(4(0(0(x1))))	(114)
4^#(0(5(1(x1))))	→	1^#(5(x1))	(187)
1^#(4(1(5(x1))))	→	1^#(2(5(1(4(x1)))))	(113)
1^#(0(3(1(x1))))	→	0^#(x1)	(111)
5^#(2(4(1(x1))))	→	5^#(x1)	(185)
5^#(4(1(5(x1))))	→	4^#(5(5(2(1(x1)))))	(107)
1^#(0(3(1(x1))))	→	1^#(0(x1))	(182)
4^#(1(0(0(x1))))	→	4^#(0(0(x1)))	(106)
5^#(4(1(5(x1))))	→	5^#(5(2(1(x1))))	(104)
1^#(2(4(3(1(x1)))))	→	4^#(x1)	(180)
5^#(4(1(x1)))	→	5^#(1(3(x1)))	(102)
1^#(0(3(1(x1))))	→	1^#(1(0(x1)))	(101)
0^#(5(4(1(1(x1)))))	→	0^#(2(5(1(1(x1)))))	(100)
5^#(0(1(x1)))	→	5^#(x1)	(176)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
1^#(4(1(5(x1))))	→	1^#(4(x1))	(99)
5^#(4(1(5(x1))))	→	1^#(x1)	(172)
4^#(0(5(1(x1))))	→	5^#(x1)	(97)
5^#(0(5(1(x1))))	→	0^#(x1)	(96)
4^#(1(0(4(x1))))	→	0^#(2(1(x1)))	(169)
1^#(0(1(4(x1))))	→	4^#(0(x1))	(95)
0^#(2(0(1(x1))))	→	0^#(0(2(1(3(x1)))))	(168)
5^#(4(2(0(1(x1)))))	→	5^#(4(0(2(1(x1)))))	(167)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)
5^#(5(0(4(x1))))	→	5^#(4(2(x1)))	(166)
5^#(5(0(4(x1))))	→	5^#(0(2(5(4(2(x1))))))	(164)
5^#(0(1(x1)))	→	5^#(5(x1))	(90)
0^#(0(1(x1)))	→	0^#(0(2(1(3(x1)))))	(88)
0^#(4(0(4(x1))))	→	4^#(4(x1))	(89)
5^#(0(1(x1)))	→	0^#(2(1(x1)))	(162)
5^#(0(1(x1)))	→	5^#(0(2(1(x1))))	(86)
5^#(4(2(0(1(x1)))))	→	4^#(0(2(1(x1))))	(160)
4^#(1(0(4(x1))))	→	4^#(4(0(2(1(x1)))))	(82)
1^#(0(1(x1)))	→	1^#(1(2(x1)))	(83)
1^#(4(1(x1)))	→	1^#(1(2(1(x1))))	(157)
5^#(0(0(1(x1))))	→	1^#(0(x1))	(81)
0^#(5(4(1(1(x1)))))	→	5^#(1(1(x1)))	(79)
5^#(0(0(1(x1))))	→	5^#(1(0(x1)))	(77)
5^#(1(0(0(5(x1)))))	→	1^#(5(x1))	(78)
5^#(4(1(5(x1))))	→	1^#(5(4(2(5(x1)))))	(155)
4^#(0(1(x1)))	→	0^#(2(1(x1)))	(74)
5^#(0(1(x1)))	→	5^#(1(x1))	(73)
5^#(1(0(0(5(x1)))))	→	5^#(0(0(2(1(5(x1))))))	(71)
5^#(1(0(0(5(x1)))))	→	0^#(2(1(5(x1))))	(68)
4^#(0(1(x1)))	→	4^#(0(2(1(x1))))	(69)
1^#(0(1(x1)))	→	0^#(2(1(1(x1))))	(67)
1^#(4(1(5(x1))))	→	5^#(1(4(x1)))	(66)
5^#(0(0(1(x1))))	→	0^#(x1)	(64)
1^#(4(1(5(x1))))	→	4^#(x1)	(65)
5^#(1(0(0(5(x1)))))	→	0^#(0(2(1(5(x1)))))	(146)
1^#(0(1(x1)))	→	0^#(2(1(1(2(x1)))))	(63)
4^#(1(0(4(x1))))	→	1^#(x1)	(144)
4^#(0(0(3(1(x1)))))	→	0^#(1(x1))	(145)
1^#(0(1(x1)))	→	1^#(1(x1))	(141)
1^#(4(1(x1)))	→	1^#(1(x1))	(58)
5^#(4(3(1(x1))))	→	5^#(4(2(1(3(x1)))))	(56)
5^#(0(1(x1)))	→	1^#(5(5(x1)))	(57)
1^#(0(1(x1)))	→	1^#(2(x1))	(138)
1^#(4(1(x1)))	→	1^#(2(1(x1)))	(136)
5^#(4(1(5(x1))))	→	5^#(4(2(5(x1))))	(55)
4^#(0(0(3(1(x1)))))	→	1^#(0(x1))	(54)
0^#(2(0(1(x1))))	→	0^#(2(1(3(x1))))	(135)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 31122
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	x₁ + 0
[4^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 31122
[5^#(x₁)]	=	x₁ + 0
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	x₁ + 0

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

0^#(1(2(0(x1))))	→	1^#(0(x1))	(196)
4^#(0(0(3(1(x1)))))	→	0^#(x1)	(127)
0^#(3(4(0(x1))))	→	0^#(3(x1))	(192)
0^#(0(1(x1)))	→	0^#(2(1(3(x1))))	(121)
4^#(0(5(1(x1))))	→	1^#(5(x1))	(187)
5^#(0(1(x1)))	→	5^#(x1)	(176)
4^#(0(5(1(x1))))	→	5^#(x1)	(97)
5^#(5(0(4(x1))))	→	5^#(4(2(x1)))	(166)
5^#(0(1(x1)))	→	5^#(5(x1))	(90)
0^#(4(0(4(x1))))	→	4^#(4(x1))	(89)
1^#(0(1(x1)))	→	1^#(1(2(x1)))	(83)
5^#(0(0(1(x1))))	→	1^#(0(x1))	(81)
0^#(5(4(1(1(x1)))))	→	5^#(1(1(x1)))	(79)
5^#(0(0(1(x1))))	→	5^#(1(0(x1)))	(77)
5^#(1(0(0(5(x1)))))	→	1^#(5(x1))	(78)
5^#(0(1(x1)))	→	5^#(1(x1))	(73)
5^#(1(0(0(5(x1)))))	→	0^#(2(1(5(x1))))	(68)
5^#(0(0(1(x1))))	→	0^#(x1)	(64)
4^#(1(0(4(x1))))	→	1^#(x1)	(144)
4^#(0(0(3(1(x1)))))	→	0^#(1(x1))	(145)
1^#(0(1(x1)))	→	1^#(1(x1))	(141)
5^#(0(1(x1)))	→	1^#(5(5(x1)))	(57)
1^#(0(1(x1)))	→	1^#(2(x1))	(138)
4^#(0(0(3(1(x1)))))	→	1^#(0(x1))	(54)
0^#(2(0(1(x1))))	→	0^#(2(1(3(x1))))	(135)

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

5^#(4(2(0(1(x1)))))	→	0^#(2(1(x1)))	(191)
5^#(4(2(0(1(x1)))))	→	4^#(0(2(1(x1))))	(160)
5^#(4(2(0(1(x1)))))	→	5^#(4(0(2(1(x1)))))	(167)
0^#(0(1(x1)))	→	0^#(0(2(1(3(x1)))))	(88)
0^#(2(0(1(x1))))	→	0^#(0(2(1(3(x1)))))	(168)
5^#(2(4(1(x1))))	→	5^#(x1)	(185)
5^#(4(1(x1)))	→	5^#(1(3(x1)))	(102)
0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
1^#(4(1(5(x1))))	→	4^#(x1)	(65)
1^#(4(1(5(x1))))	→	1^#(4(x1))	(99)
1^#(4(1(5(x1))))	→	5^#(1(4(x1)))	(66)
1^#(4(1(5(x1))))	→	1^#(2(5(1(4(x1)))))	(113)
1^#(0(1(x1)))	→	0^#(2(1(1(x1))))	(67)
1^#(4(1(x1)))	→	1^#(2(1(x1)))	(136)
1^#(4(1(x1)))	→	1^#(1(2(1(x1))))	(157)
5^#(4(3(1(x1))))	→	5^#(4(2(1(3(x1)))))	(56)
0^#(1(2(0(x1))))	→	0^#(2(1(0(x1))))	(118)
1^#(0(3(1(x1))))	→	0^#(x1)	(111)
1^#(0(3(1(x1))))	→	1^#(0(x1))	(182)
1^#(0(3(1(x1))))	→	1^#(1(0(x1)))	(101)
5^#(1(0(0(5(x1)))))	→	0^#(0(2(1(5(x1)))))	(146)
5^#(1(0(0(5(x1)))))	→	5^#(0(0(2(1(5(x1))))))	(71)
4^#(1(0(0(x1))))	→	4^#(0(0(x1)))	(106)
4^#(1(0(0(x1))))	→	1^#(4(0(0(x1))))	(114)
4^#(1(0(4(x1))))	→	0^#(2(1(x1)))	(169)
4^#(1(0(4(x1))))	→	4^#(0(2(1(x1))))	(126)
4^#(1(0(4(x1))))	→	4^#(4(0(2(1(x1)))))	(82)
5^#(0(1(x1)))	→	0^#(2(1(x1)))	(162)
5^#(0(1(x1)))	→	5^#(0(2(1(x1))))	(86)
0^#(5(4(1(1(x1)))))	→	0^#(2(5(1(1(x1)))))	(100)
0^#(5(4(1(1(x1)))))	→	4^#(0(2(5(1(1(x1))))))	(119)
1^#(0(1(4(x1))))	→	0^#(x1)	(128)
1^#(0(1(4(x1))))	→	4^#(0(x1))	(95)
1^#(0(1(4(x1))))	→	1^#(4(0(x1)))	(197)
1^#(0(1(4(x1))))	→	1^#(1(4(0(x1))))	(129)
4^#(0(1(x1)))	→	0^#(2(1(x1)))	(74)
4^#(0(1(x1)))	→	4^#(0(2(1(x1))))	(69)
1^#(4(1(x1)))	→	1^#(1(x1))	(58)
1^#(0(1(x1)))	→	0^#(2(1(1(2(x1)))))	(63)
5^#(4(1(5(x1))))	→	5^#(4(2(5(x1))))	(55)
5^#(4(1(5(x1))))	→	1^#(5(4(2(5(x1)))))	(155)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)
5^#(4(1(5(x1))))	→	1^#(x1)	(172)
5^#(4(1(5(x1))))	→	5^#(2(1(x1)))	(130)
5^#(4(1(5(x1))))	→	5^#(5(2(1(x1))))	(104)
5^#(4(1(5(x1))))	→	4^#(5(5(2(1(x1)))))	(107)
5^#(5(0(4(x1))))	→	0^#(2(5(4(2(x1)))))	(198)
5^#(5(0(4(x1))))	→	5^#(0(2(5(4(2(x1))))))	(164)
1^#(2(4(3(1(x1)))))	→	4^#(x1)	(180)
1^#(2(4(3(1(x1)))))	→	1^#(4(x1))	(190)
5^#(0(5(1(x1))))	→	0^#(x1)	(96)
5^#(0(5(1(x1))))	→	1^#(0(x1))	(116)

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)]

1	1
0	0

· x₁ +

[4(x₁)]

0	1
1	1

· x₁ +

[5(x₁)]

1	1
0	0

· x₁ +

[3(x₁)]

[4^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

[5^#(x₁)]

0	1
0	0

· x₁ +

[2(x₁)]

0	0
0	1

· x₁ +

[1^#(x₁)]

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

5^#(4(2(0(1(x1)))))	→	0^#(2(1(x1)))	(191)
5^#(4(2(0(1(x1)))))	→	4^#(0(2(1(x1))))	(160)
5^#(4(2(0(1(x1)))))	→	5^#(4(0(2(1(x1)))))	(167)
5^#(2(4(1(x1))))	→	5^#(x1)	(185)
5^#(4(1(x1)))	→	5^#(1(3(x1)))	(102)
5^#(0(1(x1)))	→	0^#(2(1(x1)))	(162)
5^#(0(1(x1)))	→	5^#(0(2(1(x1))))	(86)
5^#(4(1(5(x1))))	→	5^#(4(2(5(x1))))	(55)
5^#(4(1(5(x1))))	→	1^#(5(4(2(5(x1)))))	(155)
5^#(4(1(5(x1))))	→	1^#(x1)	(172)
5^#(4(1(5(x1))))	→	5^#(2(1(x1)))	(130)
5^#(4(1(5(x1))))	→	5^#(5(2(1(x1))))	(104)
5^#(4(1(5(x1))))	→	4^#(5(5(2(1(x1)))))	(107)
5^#(0(5(1(x1))))	→	0^#(x1)	(96)
5^#(0(5(1(x1))))	→	1^#(0(x1))	(116)

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^#(0(1(x1)))	→	0^#(0(2(1(3(x1)))))	(88)
0^#(2(0(1(x1))))	→	0^#(0(2(1(3(x1)))))	(168)
0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
1^#(4(1(5(x1))))	→	4^#(x1)	(65)
1^#(4(1(5(x1))))	→	1^#(4(x1))	(99)
1^#(4(1(5(x1))))	→	5^#(1(4(x1)))	(66)
1^#(4(1(5(x1))))	→	1^#(2(5(1(4(x1)))))	(113)
1^#(0(1(x1)))	→	0^#(2(1(1(x1))))	(67)
1^#(4(1(x1)))	→	1^#(2(1(x1)))	(136)
1^#(4(1(x1)))	→	1^#(1(2(1(x1))))	(157)
0^#(1(2(0(x1))))	→	0^#(2(1(0(x1))))	(118)
1^#(0(3(1(x1))))	→	0^#(x1)	(111)
1^#(0(3(1(x1))))	→	1^#(0(x1))	(182)
1^#(0(3(1(x1))))	→	1^#(1(0(x1)))	(101)
5^#(1(0(0(5(x1)))))	→	0^#(0(2(1(5(x1)))))	(146)
5^#(1(0(0(5(x1)))))	→	5^#(0(0(2(1(5(x1))))))	(71)
4^#(1(0(0(x1))))	→	4^#(0(0(x1)))	(106)
4^#(1(0(0(x1))))	→	1^#(4(0(0(x1))))	(114)
4^#(1(0(4(x1))))	→	0^#(2(1(x1)))	(169)
4^#(1(0(4(x1))))	→	4^#(0(2(1(x1))))	(126)
4^#(1(0(4(x1))))	→	4^#(4(0(2(1(x1)))))	(82)
0^#(5(4(1(1(x1)))))	→	0^#(2(5(1(1(x1)))))	(100)
0^#(5(4(1(1(x1)))))	→	4^#(0(2(5(1(1(x1))))))	(119)
1^#(0(1(4(x1))))	→	0^#(x1)	(128)
1^#(0(1(4(x1))))	→	4^#(0(x1))	(95)
1^#(0(1(4(x1))))	→	1^#(4(0(x1)))	(197)
1^#(0(1(4(x1))))	→	1^#(1(4(0(x1))))	(129)
4^#(0(1(x1)))	→	0^#(2(1(x1)))	(74)
4^#(0(1(x1)))	→	4^#(0(2(1(x1))))	(69)
1^#(4(1(x1)))	→	1^#(1(x1))	(58)
1^#(0(1(x1)))	→	0^#(2(1(1(2(x1)))))	(63)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)
5^#(5(0(4(x1))))	→	0^#(2(5(4(2(x1)))))	(198)
5^#(5(0(4(x1))))	→	5^#(0(2(5(4(2(x1))))))	(164)
1^#(2(4(3(1(x1)))))	→	4^#(x1)	(180)
1^#(2(4(3(1(x1)))))	→	1^#(4(x1))	(190)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	38047
[1(x₁)]	=	38047
[4(x₁)]	=	26205
[5(x₁)]	=	38048
[3(x₁)]	=	1
[4^#(x₁)]	=	38047
[0(x₁)]	=	26205
[5^#(x₁)]	=	x₁ + 0
[2(x₁)]	=	1
[1^#(x₁)]	=	38047

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

5^#(1(0(0(5(x1)))))	→	5^#(0(0(2(1(5(x1))))))	(71)
5^#(5(0(4(x1))))	→	0^#(2(5(4(2(x1)))))	(198)
5^#(5(0(4(x1))))	→	5^#(0(2(5(4(2(x1))))))	(164)

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^#(0(1(x1)))	→	0^#(0(2(1(3(x1)))))	(88)
0^#(2(0(1(x1))))	→	0^#(0(2(1(3(x1)))))	(168)
0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
1^#(4(1(5(x1))))	→	4^#(x1)	(65)
1^#(4(1(5(x1))))	→	1^#(4(x1))	(99)
1^#(4(1(5(x1))))	→	5^#(1(4(x1)))	(66)
1^#(4(1(5(x1))))	→	1^#(2(5(1(4(x1)))))	(113)
1^#(0(1(x1)))	→	0^#(2(1(1(x1))))	(67)
1^#(4(1(x1)))	→	1^#(2(1(x1)))	(136)
1^#(4(1(x1)))	→	1^#(1(2(1(x1))))	(157)
0^#(1(2(0(x1))))	→	0^#(2(1(0(x1))))	(118)
1^#(0(3(1(x1))))	→	0^#(x1)	(111)
1^#(0(3(1(x1))))	→	1^#(0(x1))	(182)
1^#(0(3(1(x1))))	→	1^#(1(0(x1)))	(101)
5^#(1(0(0(5(x1)))))	→	0^#(0(2(1(5(x1)))))	(146)
4^#(1(0(0(x1))))	→	4^#(0(0(x1)))	(106)
4^#(1(0(0(x1))))	→	1^#(4(0(0(x1))))	(114)
4^#(1(0(4(x1))))	→	0^#(2(1(x1)))	(169)
4^#(1(0(4(x1))))	→	4^#(0(2(1(x1))))	(126)
4^#(1(0(4(x1))))	→	4^#(4(0(2(1(x1)))))	(82)
0^#(5(4(1(1(x1)))))	→	0^#(2(5(1(1(x1)))))	(100)
0^#(5(4(1(1(x1)))))	→	4^#(0(2(5(1(1(x1))))))	(119)
1^#(0(1(4(x1))))	→	0^#(x1)	(128)
1^#(0(1(4(x1))))	→	4^#(0(x1))	(95)
1^#(0(1(4(x1))))	→	1^#(4(0(x1)))	(197)
1^#(0(1(4(x1))))	→	1^#(1(4(0(x1))))	(129)
4^#(0(1(x1)))	→	0^#(2(1(x1)))	(74)
4^#(0(1(x1)))	→	4^#(0(2(1(x1))))	(69)
1^#(4(1(x1)))	→	1^#(1(x1))	(58)
1^#(0(1(x1)))	→	0^#(2(1(1(2(x1)))))	(63)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)
1^#(2(4(3(1(x1)))))	→	4^#(x1)	(180)
1^#(2(4(3(1(x1)))))	→	1^#(4(x1))	(190)

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	0

· x₁ +

[1(x₁)]

0	1
0	1

· x₁ +

21092

[4(x₁)]

0	1
0	1

· x₁ +

[5(x₁)]

0	0
0	1

· x₁ +

21092

[3(x₁)]

0	0
0	1

· x₁ +

[4^#(x₁)]

0	1
0	0

· x₁ +

[0(x₁)]

0	1
0	0

· x₁ +

21091

[5^#(x₁)]

1	0
0	0

· x₁ +

21091

[2(x₁)]

1	0
0	0

· x₁ +

21091

[1^#(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

1^#(4(1(5(x1))))	→	4^#(x1)	(65)
1^#(4(1(5(x1))))	→	1^#(4(x1))	(99)
1^#(4(1(5(x1))))	→	5^#(1(4(x1)))	(66)
1^#(4(1(5(x1))))	→	1^#(2(5(1(4(x1)))))	(113)
1^#(4(1(x1)))	→	1^#(2(1(x1)))	(136)
1^#(4(1(x1)))	→	1^#(1(2(1(x1))))	(157)
0^#(1(2(0(x1))))	→	0^#(2(1(0(x1))))	(118)
1^#(0(3(1(x1))))	→	0^#(x1)	(111)
1^#(0(3(1(x1))))	→	1^#(0(x1))	(182)
1^#(0(3(1(x1))))	→	1^#(1(0(x1)))	(101)
5^#(1(0(0(5(x1)))))	→	0^#(0(2(1(5(x1)))))	(146)
4^#(1(0(0(x1))))	→	4^#(0(0(x1)))	(106)
4^#(1(0(0(x1))))	→	1^#(4(0(0(x1))))	(114)
4^#(1(0(4(x1))))	→	0^#(2(1(x1)))	(169)
4^#(1(0(4(x1))))	→	4^#(0(2(1(x1))))	(126)
4^#(1(0(4(x1))))	→	4^#(4(0(2(1(x1)))))	(82)
0^#(5(4(1(1(x1)))))	→	0^#(2(5(1(1(x1)))))	(100)
0^#(5(4(1(1(x1)))))	→	4^#(0(2(5(1(1(x1))))))	(119)
1^#(0(1(4(x1))))	→	0^#(x1)	(128)
1^#(0(1(4(x1))))	→	1^#(4(0(x1)))	(197)
1^#(0(1(4(x1))))	→	1^#(1(4(0(x1))))	(129)
1^#(4(1(x1)))	→	1^#(1(x1))	(58)
1^#(2(4(3(1(x1)))))	→	4^#(x1)	(180)
1^#(2(4(3(1(x1)))))	→	1^#(4(x1))	(190)

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^#(0(1(x1)))	→	0^#(0(2(1(3(x1)))))	(88)
0^#(2(0(1(x1))))	→	0^#(0(2(1(3(x1)))))	(168)
0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
4^#(0(1(x1)))	→	0^#(2(1(x1)))	(74)
4^#(0(1(x1)))	→	4^#(0(2(1(x1))))	(69)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	71600
[4(x₁)]	=	71600
[5(x₁)]	=	71600
[3(x₁)]	=	71600
[4^#(x₁)]	=	71600
[0(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	0
[2(x₁)]	=	43639
[1^#(x₁)]	=	0

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

0^#(0(1(x1)))	→	0^#(0(2(1(3(x1)))))	(88)
4^#(0(1(x1)))	→	0^#(2(1(x1)))	(74)

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^#(2(0(1(x1))))	→	0^#(0(2(1(3(x1)))))	(168)
0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
4^#(0(1(x1)))	→	4^#(0(2(1(x1))))	(69)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)

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)]	=	x₁ + 0
[1(x₁)]	=	16457
[4(x₁)]	=	16457
[5(x₁)]	=	16457
[3(x₁)]	=	16457
[4^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	0
[2(x₁)]	=	16456
[1^#(x₁)]	=	0

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

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

→

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

(69)

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^#(2(0(1(x1))))	→	0^#(0(2(1(3(x1)))))	(168)
0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)

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

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

0^#(3(4(0(x1))))	→	0^#(0(3(x1)))	(115)
0^#(3(4(0(x1))))	→	4^#(0(0(3(x1))))	(173)
4^#(0(0(3(1(x1)))))	→	0^#(0(1(x1)))	(165)

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

→

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

(168)

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(0^#)

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

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

and the following Max-polynomial interpretation

[1(x₁)]	=	24785
[4(x₁)]	=	17011
[5(x₁)]	=	x₁ + 7773
[3(x₁)]	=	7774
[4^#(x₁)]	=	1
[0(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	1
[2(x₁)]	=	1464
[1^#(x₁)]	=	1

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

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

→

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

(168)

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.

The 2^nd component contains the pair

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

→

1^#(3(x1))

(94)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

5(4(1(x1)))	→	4(5(2(1(3(3(x1))))))	(18)
5(4(2(0(1(x1)))))	→	2(5(4(0(2(1(x1))))))	(50)
0(4(1(x1)))	→	0(2(3(1(4(2(x1))))))	(4)
5(0(1(x1)))	→	3(0(2(3(5(1(x1))))))	(15)
1(4(0(x1)))	→	1(0(3(2(4(2(x1))))))	(8)
0(0(1(x1)))	→	0(0(2(1(3(x1)))))	(1)
0(4(1(x1)))	→	0(4(2(2(1(x1)))))	(3)
5(0(5(x1)))	→	0(3(2(5(3(5(x1))))))	(16)
0(2(0(1(x1))))	→	0(0(2(1(3(x1)))))	(21)
5(2(4(1(x1))))	→	2(1(4(2(5(x1)))))	(36)
1(2(0(4(x1))))	→	1(4(2(0(3(2(x1))))))	(26)
0(0(4(5(x1))))	→	0(5(0(2(4(2(x1))))))	(19)
5(0(0(1(x1))))	→	0(3(2(5(1(0(x1))))))	(32)
5(4(1(x1)))	→	2(4(2(5(1(3(x1))))))	(17)
1(3(0(4(x1))))	→	4(0(3(2(1(3(x1))))))	(27)
5(0(3(1(x1))))	→	0(3(5(2(2(1(x1))))))	(34)
0(3(4(0(x1))))	→	3(2(4(0(0(3(x1))))))	(22)
1(4(1(5(x1))))	→	2(1(2(5(1(4(x1))))))	(28)
0(2(4(3(1(x1)))))	→	0(2(3(1(4(2(x1))))))	(44)
1(0(1(x1)))	→	0(2(1(1(x1))))	(5)
5(0(3(1(x1))))	→	0(5(3(2(1(x1)))))	(33)
1(4(1(x1)))	→	4(2(1(1(2(1(x1))))))	(10)
5(4(3(1(x1))))	→	5(4(2(1(3(x1)))))	(39)
1(0(1(x1)))	→	1(0(3(2(1(x1)))))	(7)
0(1(2(0(x1))))	→	2(0(2(1(0(x1)))))	(20)
1(0(3(1(x1))))	→	4(2(3(1(1(0(x1))))))	(25)
5(1(0(0(5(x1)))))	→	5(0(0(2(1(5(x1))))))	(49)
4(1(0(0(x1))))	→	3(2(1(4(0(0(x1))))))	(30)
5(0(1(x1)))	→	0(3(2(1(5(5(x1))))))	(14)
4(1(0(4(x1))))	→	4(4(0(2(1(x1)))))	(31)
5(0(1(x1)))	→	5(0(2(1(x1))))	(12)
0(5(4(1(1(x1)))))	→	4(0(2(5(1(1(x1))))))	(45)
0(4(0(4(x1))))	→	0(0(3(2(4(4(x1))))))	(23)
1(0(1(4(x1))))	→	2(3(1(1(4(0(x1))))))	(24)
4(0(1(x1)))	→	4(0(2(1(x1))))	(11)
1(4(1(x1)))	→	4(2(1(1(x1))))	(9)
5(0(1(x1)))	→	0(2(1(5(3(x1)))))	(13)
5(4(3(1(x1))))	→	3(5(2(4(2(1(x1))))))	(40)
1(0(1(x1)))	→	0(2(1(1(2(x1)))))	(6)
5(4(1(5(x1))))	→	2(1(5(4(2(5(x1))))))	(38)
4(0(0(3(1(x1)))))	→	4(3(2(0(0(1(x1))))))	(48)
4(0(0(3(1(x1)))))	→	4(0(3(2(1(0(x1))))))	(47)
5(4(1(5(x1))))	→	4(5(5(2(1(x1)))))	(37)
5(4(3(1(x1))))	→	5(2(4(3(2(1(x1))))))	(41)
5(5(0(4(x1))))	→	5(0(2(5(4(2(x1))))))	(42)
1(2(4(3(1(x1)))))	→	1(2(3(3(1(4(x1))))))	(46)
5(0(5(1(x1))))	→	5(2(5(3(1(0(x1))))))	(35)
4(0(5(1(x1))))	→	0(2(4(2(1(5(x1))))))	(29)
0(0(3(3(1(x1)))))	→	0(3(0(1(2(3(x1))))))	(43)
0(0(1(x1)))	→	0(0(2(2(3(1(x1))))))	(2)

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

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

→

1^#(3(x1))

(94)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.