Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213147)

The rewrite relation of the following TRS is considered.

0(1(0(2(x1))))	→	0(0(1(3(2(x1)))))	(1)
0(1(2(2(x1))))	→	0(3(2(2(1(x1)))))	(2)
0(2(1(1(x1))))	→	0(1(1(1(3(2(x1))))))	(3)
0(2(3(2(x1))))	→	0(3(0(3(2(2(x1))))))	(4)
0(2(4(4(x1))))	→	3(0(4(2(4(x1)))))	(5)
0(4(0(2(x1))))	→	3(0(0(4(2(x1)))))	(6)
0(4(0(2(x1))))	→	3(0(0(0(4(2(x1))))))	(7)
0(4(5(1(x1))))	→	0(0(0(4(1(5(x1))))))	(8)
0(5(1(1(x1))))	→	0(1(1(3(5(x1)))))	(9)
0(5(1(1(x1))))	→	0(1(3(1(5(x1)))))	(10)
0(5(1(4(x1))))	→	0(3(3(4(1(5(x1))))))	(11)
0(5(3(2(x1))))	→	0(3(0(5(5(2(x1))))))	(12)
0(5(3(2(x1))))	→	3(0(3(5(5(2(x1))))))	(13)
0(5(4(1(x1))))	→	0(3(1(5(4(x1)))))	(14)
0(5(4(1(x1))))	→	0(0(4(1(1(5(x1))))))	(15)
0(5(4(4(x1))))	→	4(0(4(3(5(x1)))))	(16)
2(0(2(4(x1))))	→	2(2(4(0(0(x1)))))	(17)
2(0(5(1(x1))))	→	0(0(0(5(2(1(x1))))))	(18)
2(0(5(1(x1))))	→	5(0(3(5(2(1(x1))))))	(19)
2(0(5(1(x1))))	→	5(3(0(5(2(1(x1))))))	(20)
2(0(5(4(x1))))	→	2(4(0(3(5(x1)))))	(21)
5(1(0(2(x1))))	→	2(5(0(3(1(x1)))))	(22)
5(3(2(4(x1))))	→	2(3(4(3(5(x1)))))	(23)
5(4(0(2(x1))))	→	5(0(3(0(4(2(x1))))))	(24)
0(0(2(1(1(x1)))))	→	0(3(0(1(2(1(x1))))))	(25)
0(1(5(1(4(x1)))))	→	0(4(3(1(1(5(x1))))))	(26)
0(1(5(4(1(x1)))))	→	4(0(1(1(3(5(x1))))))	(27)
0(2(0(2(4(x1)))))	→	0(0(2(1(2(4(x1))))))	(28)
0(2(0(5(1(x1)))))	→	5(0(0(0(2(1(x1))))))	(29)
0(2(1(3(2(x1)))))	→	3(0(3(1(2(2(x1))))))	(30)
0(2(1(5(1(x1)))))	→	0(5(2(1(1(0(x1))))))	(31)
0(2(3(5(1(x1)))))	→	5(2(1(0(0(3(x1))))))	(32)
0(4(0(4(2(x1)))))	→	3(0(0(4(4(2(x1))))))	(33)
0(5(0(2(4(x1)))))	→	0(4(0(5(2(4(x1))))))	(34)
0(5(1(3(2(x1)))))	→	0(3(4(1(5(2(x1))))))	(35)
0(5(4(3(2(x1)))))	→	0(0(3(4(2(5(x1))))))	(36)
0(5(5(4(2(x1)))))	→	4(0(3(5(5(2(x1))))))	(37)
2(0(1(2(4(x1)))))	→	2(2(1(3(4(0(x1))))))	(38)
2(0(5(1(1(x1)))))	→	5(0(2(3(1(1(x1))))))	(39)
2(5(5(1(1(x1)))))	→	5(5(2(1(1(3(x1))))))	(40)
3(4(2(1(1(x1)))))	→	3(4(3(1(1(2(x1))))))	(41)
3(5(0(2(1(x1)))))	→	1(2(5(0(3(3(x1))))))	(42)
4(1(2(1(1(x1)))))	→	3(4(1(1(2(1(x1))))))	(43)
4(1(2(4(1(x1)))))	→	4(3(4(1(1(2(x1))))))	(44)
4(3(5(1(1(x1)))))	→	3(4(1(1(1(5(x1))))))	(45)
5(0(1(5(1(x1)))))	→	5(0(3(1(1(5(x1))))))	(46)
5(0(5(0(2(x1)))))	→	2(0(0(0(5(5(x1))))))	(47)
5(1(4(5(1(x1)))))	→	5(0(1(1(5(4(x1))))))	(48)
5(1(4(5(1(x1)))))	→	5(3(4(1(1(5(x1))))))	(49)
5(3(1(5(4(x1)))))	→	5(3(4(1(3(5(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 217 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 138 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
[2^#(x₁)]	=	x₁ + 2
[4^#(x₁)]	=	x₁ + 1
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	x₁ + 1
[2(x₁)]	=	x₁ + 3

together with the usable rules

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

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

2^#(0(5(4(x1))))	→	5^#(x1)	(256)
5^#(3(2(4(x1))))	→	5^#(x1)	(253)
0^#(4(0(2(x1))))	→	3^#(0(0(4(2(x1)))))	(254)
0^#(2(4(4(x1))))	→	3^#(0(4(2(4(x1)))))	(157)
3^#(5(0(2(1(x1)))))	→	3^#(x1)	(156)
2^#(0(1(2(4(x1)))))	→	0^#(x1)	(155)
0^#(2(1(1(x1))))	→	3^#(2(x1))	(248)
0^#(2(3(2(x1))))	→	3^#(2(2(x1)))	(148)
0^#(5(1(3(2(x1)))))	→	3^#(4(1(5(2(x1)))))	(147)
0^#(2(4(4(x1))))	→	2^#(4(x1))	(144)
2^#(0(5(4(x1))))	→	0^#(3(5(x1)))	(244)
5^#(3(2(4(x1))))	→	3^#(5(x1))	(141)
5^#(3(1(5(4(x1)))))	→	3^#(4(1(3(5(x1)))))	(136)
0^#(2(3(2(x1))))	→	3^#(0(3(2(2(x1)))))	(236)
0^#(2(3(5(1(x1)))))	→	0^#(0(3(x1)))	(237)
5^#(3(2(4(x1))))	→	4^#(3(5(x1)))	(130)
3^#(5(0(2(1(x1)))))	→	0^#(3(3(x1)))	(127)
5^#(3(2(4(x1))))	→	3^#(4(3(5(x1))))	(230)
3^#(5(0(2(1(x1)))))	→	3^#(3(x1))	(124)
0^#(2(1(1(x1))))	→	2^#(x1)	(121)
3^#(5(0(2(1(x1)))))	→	2^#(5(0(3(3(x1)))))	(226)
5^#(3(2(4(x1))))	→	2^#(3(4(3(5(x1)))))	(120)
5^#(0(5(0(2(x1)))))	→	0^#(0(0(5(5(x1)))))	(225)
2^#(0(2(4(x1))))	→	0^#(0(x1))	(117)
2^#(5(5(1(1(x1)))))	→	3^#(x1)	(118)
0^#(5(1(4(x1))))	→	3^#(4(1(5(x1))))	(223)
2^#(0(5(4(x1))))	→	3^#(5(x1))	(220)
0^#(5(3(2(x1))))	→	3^#(0(5(5(2(x1)))))	(219)
0^#(2(1(3(2(x1)))))	→	2^#(2(x1))	(218)
0^#(5(4(4(x1))))	→	3^#(5(x1))	(212)
0^#(5(4(3(2(x1)))))	→	2^#(5(x1))	(213)
5^#(3(1(5(4(x1)))))	→	3^#(5(x1))	(210)
0^#(5(5(4(2(x1)))))	→	3^#(5(5(2(x1))))	(207)
5^#(0(5(0(2(x1)))))	→	5^#(x1)	(107)
0^#(5(1(4(x1))))	→	3^#(3(4(1(5(x1)))))	(105)
0^#(1(0(2(x1))))	→	3^#(2(x1))	(101)
0^#(5(3(2(x1))))	→	3^#(5(5(2(x1))))	(202)
0^#(2(3(5(1(x1)))))	→	0^#(3(x1))	(199)
2^#(0(2(4(x1))))	→	4^#(0(0(x1)))	(96)
4^#(1(2(4(1(x1)))))	→	2^#(x1)	(196)
2^#(0(2(4(x1))))	→	2^#(4(0(0(x1))))	(94)
0^#(1(5(4(1(x1)))))	→	3^#(5(x1))	(93)
2^#(0(2(4(x1))))	→	0^#(x1)	(194)
5^#(0(5(0(2(x1)))))	→	0^#(5(5(x1)))	(92)
0^#(5(3(2(x1))))	→	3^#(0(3(5(5(2(x1))))))	(87)
2^#(0(1(2(4(x1)))))	→	4^#(0(x1))	(80)
3^#(5(0(2(1(x1)))))	→	5^#(0(3(3(x1))))	(185)
5^#(4(0(2(x1))))	→	3^#(0(4(2(x1))))	(72)
2^#(0(1(2(4(x1)))))	→	3^#(4(0(x1)))	(73)
0^#(2(3(5(1(x1)))))	→	3^#(x1)	(177)
0^#(4(0(2(x1))))	→	3^#(0(0(0(4(2(x1))))))	(67)
5^#(0(5(0(2(x1)))))	→	0^#(0(5(5(x1))))	(66)
0^#(5(1(1(x1))))	→	3^#(5(x1))	(174)
0^#(2(1(5(1(x1)))))	→	0^#(x1)	(170)
0^#(5(4(3(2(x1)))))	→	5^#(x1)	(58)
0^#(2(3(2(x1))))	→	2^#(2(x1))	(165)
5^#(0(5(0(2(x1)))))	→	5^#(5(x1))	(166)
5^#(0(5(0(2(x1)))))	→	2^#(0(0(0(5(5(x1))))))	(55)
0^#(5(4(3(2(x1)))))	→	3^#(4(2(5(x1))))	(163)
3^#(4(2(1(1(x1)))))	→	2^#(x1)	(54)
2^#(0(5(4(x1))))	→	4^#(0(3(5(x1))))	(161)
0^#(4(0(4(2(x1)))))	→	3^#(0(0(4(4(2(x1))))))	(162)

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

2^#(0(5(4(x1))))	→	2^#(4(0(3(5(x1)))))	(83)
2^#(0(2(4(x1))))	→	2^#(2(4(0(0(x1)))))	(167)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

2^#(0(5(4(x1))))	→	2^#(4(0(3(5(x1)))))	(83)
2^#(0(2(4(x1))))	→	2^#(2(4(0(0(x1)))))	(167)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(2(3(2(x1))))	→	0^#(3(2(2(x1))))	(239)
0^#(2(3(2(x1))))	→	0^#(3(0(3(2(2(x1))))))	(133)
0^#(5(4(1(x1))))	→	5^#(x1)	(113)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(4(5(1(x1))))	→	0^#(4(1(5(x1))))	(98)
0^#(4(5(1(x1))))	→	0^#(0(4(1(5(x1)))))	(235)
0^#(4(5(1(x1))))	→	0^#(0(0(4(1(5(x1))))))	(193)
0^#(1(0(2(x1))))	→	0^#(1(3(2(x1))))	(211)
0^#(1(0(2(x1))))	→	0^#(0(1(3(2(x1)))))	(74)
0^#(5(4(4(x1))))	→	5^#(x1)	(200)
0^#(5(4(4(x1))))	→	4^#(3(5(x1)))	(150)
0^#(5(4(4(x1))))	→	0^#(4(3(5(x1))))	(178)
0^#(5(4(4(x1))))	→	4^#(0(4(3(5(x1)))))	(190)
0^#(5(4(3(2(x1)))))	→	4^#(2(5(x1)))	(171)
0^#(5(4(3(2(x1)))))	→	0^#(3(4(2(5(x1)))))	(76)
0^#(5(4(3(2(x1)))))	→	0^#(0(3(4(2(5(x1))))))	(205)
0^#(1(5(1(4(x1)))))	→	5^#(x1)	(189)
0^#(1(5(4(1(x1)))))	→	5^#(x1)	(240)
0^#(5(0(2(4(x1)))))	→	5^#(2(4(x1)))	(183)
0^#(5(0(2(4(x1)))))	→	0^#(5(2(4(x1))))	(78)
0^#(5(0(2(4(x1)))))	→	4^#(0(5(2(4(x1)))))	(109)
0^#(5(0(2(4(x1)))))	→	0^#(4(0(5(2(4(x1))))))	(246)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(2(4(4(x1))))	→	4^#(2(4(x1)))	(103)
0^#(2(4(4(x1))))	→	0^#(4(2(4(x1))))	(224)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(4(2(x1)))))	→	0^#(4(4(2(x1))))	(128)
0^#(4(0(4(2(x1)))))	→	0^#(0(4(4(2(x1)))))	(186)
0^#(5(1(1(x1))))	→	5^#(x1)	(182)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
0^#(4(0(2(x1))))	→	0^#(0(0(4(2(x1)))))	(61)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(4(1(x1))))	→	4^#(x1)	(132)
0^#(5(4(1(x1))))	→	5^#(4(x1))	(64)
0^#(5(3(2(x1))))	→	5^#(2(x1))	(151)
0^#(5(3(2(x1))))	→	5^#(5(2(x1)))	(91)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
0^#(5(3(2(x1))))	→	0^#(3(0(5(5(2(x1))))))	(191)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	0^#(3(0(4(2(x1)))))	(176)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(5(1(4(x1))))	→	5^#(x1)	(126)
0^#(5(1(4(x1))))	→	0^#(3(3(4(1(5(x1))))))	(57)
0^#(5(1(1(x1))))	→	5^#(x1)	(182)
0^#(5(3(2(x1))))	→	5^#(2(x1))	(151)
0^#(5(3(2(x1))))	→	5^#(5(2(x1)))	(91)
0^#(5(3(2(x1))))	→	0^#(3(5(5(2(x1)))))	(100)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
0^#(5(5(4(2(x1)))))	→	5^#(2(x1))	(188)
0^#(5(5(4(2(x1)))))	→	5^#(5(2(x1)))	(175)
0^#(5(5(4(2(x1)))))	→	0^#(3(5(5(2(x1)))))	(139)
0^#(5(5(4(2(x1)))))	→	4^#(0(3(5(5(2(x1))))))	(152)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(5(1(3(2(x1)))))	→	5^#(2(x1))	(227)
0^#(5(1(3(2(x1)))))	→	0^#(3(4(1(5(2(x1))))))	(111)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)
0^#(2(0(5(1(x1)))))	→	5^#(0(0(0(2(1(x1))))))	(238)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

0^#(5(4(1(x1))))	→	5^#(x1)	(113)
0^#(5(4(4(x1))))	→	5^#(x1)	(200)
0^#(5(4(4(x1))))	→	4^#(3(5(x1)))	(150)
0^#(5(4(4(x1))))	→	0^#(4(3(5(x1))))	(178)
0^#(5(4(4(x1))))	→	4^#(0(4(3(5(x1)))))	(190)
0^#(5(4(3(2(x1)))))	→	4^#(2(5(x1)))	(171)
0^#(5(4(3(2(x1)))))	→	0^#(3(4(2(5(x1)))))	(76)
0^#(5(4(3(2(x1)))))	→	0^#(0(3(4(2(5(x1))))))	(205)
0^#(1(5(1(4(x1)))))	→	5^#(x1)	(189)
0^#(1(5(4(1(x1)))))	→	5^#(x1)	(240)
0^#(5(0(2(4(x1)))))	→	5^#(2(4(x1)))	(183)
0^#(5(0(2(4(x1)))))	→	4^#(0(5(2(4(x1)))))	(109)
0^#(5(0(2(4(x1)))))	→	0^#(4(0(5(2(4(x1))))))	(246)
0^#(2(4(4(x1))))	→	4^#(2(4(x1)))	(103)
0^#(2(4(4(x1))))	→	0^#(4(2(4(x1))))	(224)
0^#(5(1(1(x1))))	→	5^#(x1)	(182)
0^#(5(4(1(x1))))	→	4^#(x1)	(132)
0^#(5(4(1(x1))))	→	5^#(4(x1))	(64)
0^#(5(3(2(x1))))	→	5^#(2(x1))	(151)
0^#(5(3(2(x1))))	→	5^#(5(2(x1)))	(91)
0^#(5(1(4(x1))))	→	5^#(x1)	(126)
0^#(5(1(4(x1))))	→	0^#(3(3(4(1(5(x1))))))	(57)
0^#(5(1(1(x1))))	→	5^#(x1)	(182)
0^#(5(3(2(x1))))	→	5^#(2(x1))	(151)
0^#(5(3(2(x1))))	→	5^#(5(2(x1)))	(91)
0^#(5(5(4(2(x1)))))	→	5^#(2(x1))	(188)
0^#(5(5(4(2(x1)))))	→	5^#(5(2(x1)))	(175)
0^#(5(5(4(2(x1)))))	→	4^#(0(3(5(5(2(x1))))))	(152)
0^#(5(1(3(2(x1)))))	→	5^#(2(x1))	(227)
0^#(5(1(3(2(x1)))))	→	0^#(3(4(1(5(2(x1))))))	(111)
0^#(2(0(5(1(x1)))))	→	5^#(0(0(0(2(1(x1))))))	(238)

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

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(2(3(2(x1))))	→	0^#(3(2(2(x1))))	(239)
0^#(2(3(2(x1))))	→	0^#(3(0(3(2(2(x1))))))	(133)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(4(5(1(x1))))	→	0^#(4(1(5(x1))))	(98)
0^#(4(5(1(x1))))	→	0^#(0(4(1(5(x1)))))	(235)
0^#(4(5(1(x1))))	→	0^#(0(0(4(1(5(x1))))))	(193)
0^#(1(0(2(x1))))	→	0^#(1(3(2(x1))))	(211)
0^#(1(0(2(x1))))	→	0^#(0(1(3(2(x1)))))	(74)
0^#(5(0(2(4(x1)))))	→	0^#(5(2(4(x1))))	(78)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(4(2(x1)))))	→	0^#(4(4(2(x1))))	(128)
0^#(4(0(4(2(x1)))))	→	0^#(0(4(4(2(x1)))))	(186)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
0^#(4(0(2(x1))))	→	0^#(0(0(4(2(x1)))))	(61)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
0^#(5(3(2(x1))))	→	0^#(3(0(5(5(2(x1))))))	(191)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	0^#(3(0(4(2(x1)))))	(176)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(5(3(2(x1))))	→	0^#(3(5(5(2(x1)))))	(100)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
0^#(5(5(4(2(x1)))))	→	0^#(3(5(5(2(x1)))))	(139)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	1
[4(x₁)]	=	1
[5(x₁)]	=	26954
[3(x₁)]	=	1
[2^#(x₁)]	=	2
[4^#(x₁)]	=	1
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	0
[5^#(x₁)]	=	1
[2(x₁)]	=	26954

together with the usable rules

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

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

0^#(2(3(2(x1))))	→	0^#(3(2(2(x1))))	(239)
0^#(2(3(2(x1))))	→	0^#(3(0(3(2(2(x1))))))	(133)
0^#(5(3(2(x1))))	→	0^#(3(0(5(5(2(x1))))))	(191)
0^#(5(3(2(x1))))	→	0^#(3(5(5(2(x1)))))	(100)
0^#(5(5(4(2(x1)))))	→	0^#(3(5(5(2(x1)))))	(139)

could be deleted.

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

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(4(5(1(x1))))	→	0^#(4(1(5(x1))))	(98)
0^#(4(5(1(x1))))	→	0^#(0(4(1(5(x1)))))	(235)
0^#(4(5(1(x1))))	→	0^#(0(0(4(1(5(x1))))))	(193)
0^#(1(0(2(x1))))	→	0^#(1(3(2(x1))))	(211)
0^#(1(0(2(x1))))	→	0^#(0(1(3(2(x1)))))	(74)
0^#(5(0(2(4(x1)))))	→	0^#(5(2(4(x1))))	(78)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(4(2(x1)))))	→	0^#(4(4(2(x1))))	(128)
0^#(4(0(4(2(x1)))))	→	0^#(0(4(4(2(x1)))))	(186)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
0^#(4(0(2(x1))))	→	0^#(0(0(4(2(x1)))))	(61)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	0^#(3(0(4(2(x1)))))	(176)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

1.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	1
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	26325
[3(x₁)]	=	1
[2^#(x₁)]	=	2
[4^#(x₁)]	=	26325
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	0
[5^#(x₁)]	=	26325
[2(x₁)]	=	26325

together with the usable rules

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

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

0^#(4(5(1(x1))))	→	0^#(4(1(5(x1))))	(98)
0^#(4(5(1(x1))))	→	0^#(0(4(1(5(x1)))))	(235)
0^#(4(5(1(x1))))	→	0^#(0(0(4(1(5(x1))))))	(193)
5^#(4(0(2(x1))))	→	0^#(3(0(4(2(x1)))))	(176)

could be deleted.

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

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(1(0(2(x1))))	→	0^#(1(3(2(x1))))	(211)
0^#(1(0(2(x1))))	→	0^#(0(1(3(2(x1)))))	(74)
0^#(5(0(2(4(x1)))))	→	0^#(5(2(4(x1))))	(78)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(4(2(x1)))))	→	0^#(4(4(2(x1))))	(128)
0^#(4(0(4(2(x1)))))	→	0^#(0(4(4(2(x1)))))	(186)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
0^#(4(0(2(x1))))	→	0^#(0(0(4(2(x1)))))	(61)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

1.1.1.1.2.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(0^#)	=	1
π(4)	=	1

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

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

and the following Max-polynomial interpretation

[1(x₁)]	=	10113
[5(x₁)]	=	10114
[3(x₁)]	=	10113
[2^#(x₁)]	=	1
[4^#(x₁)]	=	10114
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	1
[5^#(x₁)]	=	10114
[2(x₁)]	=	10114

together with the usable rules

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

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

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

→

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

(74)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(5(0(2(4(x1)))))	→	0^#(5(2(4(x1))))	(78)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(4(2(x1)))))	→	0^#(4(4(2(x1))))	(128)
0^#(4(0(4(2(x1)))))	→	0^#(0(4(4(2(x1)))))	(186)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
0^#(4(0(2(x1))))	→	0^#(0(0(4(2(x1)))))	(61)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

1.1.1.1.2.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(0^#)	=	1
π(3)	=	1

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

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

and the following Max-polynomial interpretation

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

together with the usable rules

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

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

0^#(4(0(4(2(x1)))))	→	0^#(0(4(4(2(x1)))))	(186)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)
0^#(4(0(2(x1))))	→	0^#(0(0(4(2(x1)))))	(61)
0^#(4(0(2(x1))))	→	0^#(0(4(2(x1))))	(95)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(5(0(2(4(x1)))))	→	0^#(5(2(4(x1))))	(78)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(4(2(x1)))))	→	0^#(4(4(2(x1))))	(128)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1 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)]

0	1
0	0

· x₁ +

[5(x₁)]

12052

12051

[3(x₁)]

1	0
0	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

12052

[0(x₁)]

12052

12051

[3^#(x₁)]

[5^#(x₁)]

12052

[2(x₁)]

0	1
0	0

· x₁ +

12051

together with the usable rules

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

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

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

→

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

(128)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(5(0(2(4(x1)))))	→	0^#(5(2(4(x1))))	(78)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1 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	0

· x₁ +

11634

[1(x₁)]

0	1
0	0

· x₁ +

418

[4(x₁)]

418

[5(x₁)]

0	0
0	1

· x₁ +

418

[3(x₁)]

x₁ +

[2^#(x₁)]

[4^#(x₁)]

12052

[0(x₁)]

418

[3^#(x₁)]

[5^#(x₁)]

12052

[2(x₁)]

0	0
1	0

· x₁ +

418

together with the usable rules

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

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

0^#(5(0(2(4(x1)))))

→

0^#(5(2(4(x1))))

(78)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)
0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1 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 +

11635

[1(x₁)]

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

0	1
0	0

· x₁ +

[3(x₁)]

[2^#(x₁)]

[4^#(x₁)]

11635

[0(x₁)]

1	0
0	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

11635

[2(x₁)]

1	0
1	0

· x₁ +

together with the usable rules

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

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

0^#(2(0(5(1(x1)))))	→	0^#(0(2(1(x1))))	(169)
0^#(2(0(5(1(x1)))))	→	0^#(0(0(2(1(x1)))))	(112)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(2(0(2(4(x1)))))	→	0^#(0(2(1(2(4(x1))))))	(88)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(0^#)	=	1
π(0)	=	1
π(2)	=	1

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

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

and the following Max-polynomial interpretation

[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	6694
[5(x₁)]	=	6695
[3(x₁)]	=	x₁ + 0
[2^#(x₁)]	=	0
[4^#(x₁)]	=	6694
[3^#(x₁)]	=	1
[5^#(x₁)]	=	6694

together with the usable rules

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

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

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

→

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

(88)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

0	1
1	1

· x₁ +

2456

[1(x₁)]

9178

[4(x₁)]

1	0
1	1

· x₁ +

[5(x₁)]

0	1
0	0

· x₁ +

[3(x₁)]

0	0
1	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

11635

18357

[0(x₁)]

1	0
1	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

11635

18357

[2(x₁)]

9178

together with the usable rules

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

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

0^#(4(0(4(2(x1)))))	→	4^#(4(2(x1)))	(159)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)
0^#(4(0(2(x1))))	→	4^#(2(x1))	(75)
0^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(154)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
0^#(5(3(2(x1))))	→	0^#(5(5(2(x1))))	(52)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

18357

[1(x₁)]

0	0
1	1

· x₁ +

[4(x₁)]

0	0
1	1

· x₁ +

[5(x₁)]

0	1
0	0

· x₁ +

[3(x₁)]

1	0
1	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

18357

[0(x₁)]

0	0
1	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

18357

[2(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

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

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

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

→

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

(52)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
0^#(4(5(1(x1))))	→	5^#(x1)	(62)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	4^#(2(x1))	(82)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

18357

[1(x₁)]

1	0
1	1

· x₁ +

11166

[4(x₁)]

x₁ +

[5(x₁)]

0	1
0	0

· x₁ +

[3(x₁)]

0	0
1	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

0	1
0	0

· x₁ +

11165

18357

[0(x₁)]

0	0
1	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

0	1
0	0

· x₁ +

11165

18357

[2(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

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

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

0^#(4(5(1(x1))))	→	5^#(x1)	(62)
5^#(1(4(5(1(x1)))))	→	5^#(x1)	(252)
4^#(3(5(1(1(x1)))))	→	5^#(x1)	(56)
5^#(4(0(2(x1))))	→	0^#(4(2(x1)))	(85)
5^#(1(4(5(1(x1)))))	→	4^#(x1)	(181)
5^#(1(4(5(1(x1)))))	→	5^#(4(x1))	(53)
5^#(0(1(5(1(x1)))))	→	5^#(x1)	(160)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)
5^#(4(0(2(x1))))	→	5^#(0(3(0(4(2(x1))))))	(77)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

18357

[1(x₁)]

x₁ +

[4(x₁)]

0	0
1	0

· x₁ +

8637

[5(x₁)]

1	1
0	1

· x₁ +

[3(x₁)]

1	0
0	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

11165

18357

[0(x₁)]

0	0
1	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

1	0
0	0

· x₁ +

11165

18357

[2(x₁)]

together with the usable rules

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

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

5^#(3(1(5(4(x1)))))	→	5^#(x1)	(192)
5^#(3(1(5(4(x1)))))	→	5^#(3(4(1(3(5(x1))))))	(203)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(4(0(2(x1))))

→

5^#(0(3(0(4(2(x1))))))

(77)

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

5^#(4(0(2(x1))))

→

5^#(0(3(0(4(2(x1))))))

(77)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

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

→

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

(211)

1.1.1.1.2.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

0	1
0	0

· x₁ +

18357

[1(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

[5(x₁)]

0	1
0	0

· x₁ +

[3(x₁)]

0	0
1	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

11165

18357

[0(x₁)]

x₁ +

[3^#(x₁)]

[5^#(x₁)]

11165

18357

[2(x₁)]

together with the usable rules

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

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

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

→

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

(211)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.