Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/214183)

The rewrite relation of the following TRS is considered.

0(0(1(2(x1))))	→	2(3(1(0(0(x1)))))	(1)
0(0(1(2(x1))))	→	3(1(0(0(2(x1)))))	(2)
0(0(1(2(x1))))	→	0(0(2(3(1(1(x1))))))	(3)
0(1(0(4(x1))))	→	3(1(4(0(0(x1)))))	(4)
0(1(0(4(x1))))	→	3(4(0(0(3(1(x1))))))	(5)
0(1(0(4(x1))))	→	4(3(1(0(5(0(x1))))))	(6)
0(1(0(4(x1))))	→	5(0(4(3(1(0(x1))))))	(7)
0(1(0(5(x1))))	→	3(5(3(1(0(0(x1))))))	(8)
0(1(0(5(x1))))	→	5(3(1(0(0(2(x1))))))	(9)
0(1(4(2(x1))))	→	2(3(1(4(0(x1)))))	(10)
0(1(4(2(x1))))	→	4(0(3(1(2(4(x1))))))	(11)
0(1(4(2(x1))))	→	4(3(1(1(0(2(x1))))))	(12)
0(5(0(4(x1))))	→	4(0(0(3(5(x1)))))	(13)
0(5(0(4(x1))))	→	5(0(0(3(4(x1)))))	(14)
0(5(0(5(x1))))	→	3(5(2(0(0(5(x1))))))	(15)
0(5(0(5(x1))))	→	5(3(5(4(0(0(x1))))))	(16)
0(5(0(5(x1))))	→	5(5(3(4(0(0(x1))))))	(17)
2(0(1(2(x1))))	→	0(2(4(1(2(x1)))))	(18)
2(0(1(2(x1))))	→	2(5(0(2(2(1(x1))))))	(19)
2(0(1(2(x1))))	→	3(1(3(2(2(0(x1))))))	(20)
2(0(4(2(x1))))	→	4(0(2(2(1(x1)))))	(21)
2(0(4(2(x1))))	→	4(0(2(1(2(1(x1))))))	(22)
2(0(5(2(x1))))	→	0(2(1(5(2(2(x1))))))	(23)
3(0(4(2(x1))))	→	0(3(4(1(1(2(x1))))))	(24)
3(0(4(2(x1))))	→	4(3(3(1(0(2(x1))))))	(25)
0(0(1(2(2(x1)))))	→	0(0(2(1(2(1(x1))))))	(26)
0(0(1(3(2(x1)))))	→	0(3(1(2(0(4(x1))))))	(27)
0(0(1(3(2(x1)))))	→	2(5(0(3(1(0(x1))))))	(28)
0(0(5(2(2(x1)))))	→	0(0(2(5(2(1(x1))))))	(29)
0(1(0(5(2(x1)))))	→	0(2(1(0(3(5(x1))))))	(30)
0(1(0(5(5(x1)))))	→	1(3(5(0(0(5(x1))))))	(31)
0(1(2(0(5(x1)))))	→	4(0(5(2(1(0(x1))))))	(32)
0(1(3(0(4(x1)))))	→	1(0(3(4(4(0(x1))))))	(33)
0(1(3(4(2(x1)))))	→	4(2(1(0(3(4(x1))))))	(34)
0(1(4(2(2(x1)))))	→	1(1(0(2(4(2(x1))))))	(35)
0(1(4(2(4(x1)))))	→	4(0(2(3(1(4(x1))))))	(36)
0(1(4(3(2(x1)))))	→	3(1(2(0(4(4(x1))))))	(37)
0(3(0(0(5(x1)))))	→	5(3(1(0(0(0(x1))))))	(38)
0(4(1(0(4(x1)))))	→	4(0(2(1(4(0(x1))))))	(39)
0(4(3(2(2(x1)))))	→	4(2(0(3(5(2(x1))))))	(40)
0(5(1(0(4(x1)))))	→	5(2(4(1(0(0(x1))))))	(41)
0(5(1(4(2(x1)))))	→	4(0(3(5(1(2(x1))))))	(42)
2(1(0(1(2(x1)))))	→	2(1(1(0(2(3(x1))))))	(43)
2(4(0(5(2(x1)))))	→	0(4(2(3(5(2(x1))))))	(44)
3(0(0(1(2(x1)))))	→	0(3(5(1(0(2(x1))))))	(45)
3(0(0(5(2(x1)))))	→	2(0(0(3(5(1(x1))))))	(46)
3(0(5(4(2(x1)))))	→	4(0(3(5(2(1(x1))))))	(47)
3(2(0(1(2(x1)))))	→	2(1(3(1(0(2(x1))))))	(48)
3(2(0(5(2(x1)))))	→	2(0(2(3(4(5(x1))))))	(49)
3(2(0(5(2(x1)))))	→	5(3(0(2(2(2(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 163 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(0(1(2(2(x1)))))	→	2^#(1(2(1(x1))))	(138)
2^#(0(4(2(x1))))	→	2^#(1(2(1(x1))))	(137)
2^#(0(1(2(x1))))	→	3^#(2(2(0(x1))))	(206)
0^#(0(1(2(2(x1)))))	→	2^#(1(x1))	(136)
0^#(0(1(3(2(x1)))))	→	0^#(x1)	(130)
2^#(0(1(2(x1))))	→	2^#(0(x1))	(127)
0^#(4(1(0(4(x1)))))	→	0^#(x1)	(125)
2^#(0(5(2(x1))))	→	2^#(2(x1))	(124)
3^#(2(0(1(2(x1)))))	→	0^#(2(x1))	(122)
0^#(1(2(0(5(x1)))))	→	2^#(1(0(x1)))	(199)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
3^#(2(0(5(2(x1)))))	→	2^#(2(2(x1)))	(195)
3^#(0(4(2(x1))))	→	0^#(2(x1))	(116)
2^#(0(1(2(x1))))	→	0^#(2(2(1(x1))))	(193)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(0(1(2(2(x1)))))	→	0^#(2(1(2(1(x1)))))	(114)
2^#(0(1(2(x1))))	→	2^#(2(0(x1)))	(115)
3^#(2(0(5(2(x1)))))	→	0^#(2(2(2(x1))))	(189)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
0^#(0(1(2(x1))))	→	0^#(2(x1))	(188)
2^#(0(4(2(x1))))	→	2^#(1(x1))	(108)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(1(2(0(5(x1)))))	→	0^#(x1)	(182)
0^#(0(5(2(2(x1)))))	→	2^#(1(x1))	(103)
2^#(0(4(2(x1))))	→	2^#(2(1(x1)))	(177)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
2^#(0(1(2(x1))))	→	2^#(2(1(x1)))	(175)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
3^#(0(0(1(2(x1)))))	→	0^#(2(x1))	(91)
3^#(2(0(5(2(x1)))))	→	2^#(2(x1))	(167)
2^#(1(0(1(2(x1)))))	→	2^#(3(x1))	(90)
0^#(0(1(3(2(x1)))))	→	0^#(4(x1))	(88)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(1(0(5(x1))))	→	0^#(x1)	(86)
0^#(1(0(4(x1))))	→	0^#(x1)	(52)
3^#(2(0(5(2(x1)))))	→	3^#(0(2(2(2(x1)))))	(84)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(4(x1))))	→	0^#(x1)	(52)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(3(0(0(5(x1)))))	→	0^#(0(x1))	(160)
0^#(1(0(5(x1))))	→	2^#(x1)	(76)
0^#(0(1(2(x1))))	→	0^#(x1)	(74)
0^#(5(0(5(x1))))	→	0^#(x1)	(75)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
0^#(5(1(0(4(x1)))))	→	0^#(x1)	(158)
3^#(0(5(4(2(x1)))))	→	2^#(1(x1))	(157)
0^#(1(0(5(x1))))	→	0^#(2(x1))	(71)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
2^#(1(0(1(2(x1)))))	→	0^#(2(3(x1)))	(153)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(1(0(5(5(x1)))))	→	0^#(5(x1))	(148)
0^#(3(0(0(5(x1)))))	→	0^#(0(0(x1)))	(149)
0^#(5(0(5(x1))))	→	0^#(x1)	(75)
0^#(1(3(0(4(x1)))))	→	0^#(x1)	(62)
0^#(0(1(3(2(x1)))))	→	2^#(0(4(x1)))	(60)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
2^#(0(4(2(x1))))	→	2^#(1(x1))	(108)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
2^#(0(1(2(x1))))	→	0^#(x1)	(56)
2^#(1(0(1(2(x1)))))	→	3^#(x1)	(142)
0^#(3(0(0(5(x1)))))	→	0^#(x1)	(55)
2^#(0(1(2(x1))))	→	2^#(1(x1))	(140)
0^#(1(0(4(x1))))	→	0^#(x1)	(52)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 38145
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	x₁ + 0
[2^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 38145
[3^#(x₁)]	=	x₁ + 0
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

0^#(0(1(2(2(x1)))))	→	2^#(1(2(1(x1))))	(138)
2^#(0(4(2(x1))))	→	2^#(1(2(1(x1))))	(137)
0^#(0(1(2(2(x1)))))	→	2^#(1(x1))	(136)
0^#(0(1(3(2(x1)))))	→	0^#(x1)	(130)
0^#(4(1(0(4(x1)))))	→	0^#(x1)	(125)
2^#(0(5(2(x1))))	→	2^#(2(x1))	(124)
0^#(1(2(0(5(x1)))))	→	2^#(1(0(x1)))	(199)
3^#(2(0(5(2(x1)))))	→	2^#(2(2(x1)))	(195)
0^#(0(1(2(2(x1)))))	→	0^#(2(1(2(1(x1)))))	(114)
0^#(0(1(2(x1))))	→	0^#(2(x1))	(188)
2^#(0(4(2(x1))))	→	2^#(1(x1))	(108)
0^#(1(2(0(5(x1)))))	→	0^#(x1)	(182)
0^#(0(5(2(2(x1)))))	→	2^#(1(x1))	(103)
2^#(0(4(2(x1))))	→	2^#(2(1(x1)))	(177)
2^#(0(1(2(x1))))	→	2^#(2(1(x1)))	(175)
3^#(0(0(1(2(x1)))))	→	0^#(2(x1))	(91)
3^#(2(0(5(2(x1)))))	→	2^#(2(x1))	(167)
2^#(1(0(1(2(x1)))))	→	2^#(3(x1))	(90)
0^#(0(1(3(2(x1)))))	→	0^#(4(x1))	(88)
0^#(1(0(5(x1))))	→	0^#(x1)	(86)
0^#(1(0(4(x1))))	→	0^#(x1)	(52)
0^#(1(0(4(x1))))	→	0^#(x1)	(52)
0^#(3(0(0(5(x1)))))	→	0^#(0(x1))	(160)
0^#(1(0(5(x1))))	→	2^#(x1)	(76)
0^#(0(1(2(x1))))	→	0^#(x1)	(74)
0^#(5(0(5(x1))))	→	0^#(x1)	(75)
0^#(5(1(0(4(x1)))))	→	0^#(x1)	(158)
3^#(0(5(4(2(x1)))))	→	2^#(1(x1))	(157)
0^#(1(0(5(x1))))	→	0^#(2(x1))	(71)
0^#(1(0(5(5(x1)))))	→	0^#(5(x1))	(148)
0^#(5(0(5(x1))))	→	0^#(x1)	(75)
0^#(1(3(0(4(x1)))))	→	0^#(x1)	(62)
0^#(0(1(3(2(x1)))))	→	2^#(0(4(x1)))	(60)
2^#(0(4(2(x1))))	→	2^#(1(x1))	(108)
2^#(1(0(1(2(x1)))))	→	3^#(x1)	(142)
0^#(3(0(0(5(x1)))))	→	0^#(x1)	(55)
2^#(0(1(2(x1))))	→	2^#(1(x1))	(140)
0^#(1(0(4(x1))))	→	0^#(x1)	(52)

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

3^#(2(0(5(2(x1)))))	→	0^#(2(2(2(x1))))	(189)
3^#(2(0(5(2(x1)))))	→	3^#(0(2(2(2(x1)))))	(84)
0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
2^#(0(1(2(x1))))	→	0^#(2(2(1(x1))))	(193)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
2^#(0(1(2(x1))))	→	0^#(x1)	(56)
2^#(0(1(2(x1))))	→	2^#(0(x1))	(127)
2^#(0(1(2(x1))))	→	2^#(2(0(x1)))	(115)
2^#(0(1(2(x1))))	→	3^#(2(2(0(x1))))	(206)
3^#(0(4(2(x1))))	→	0^#(2(x1))	(116)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(3(0(0(5(x1)))))	→	0^#(0(0(x1)))	(149)
3^#(2(0(1(2(x1)))))	→	0^#(2(x1))	(122)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
2^#(1(0(1(2(x1)))))	→	0^#(2(3(x1)))	(153)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

[3(x₁)]

1	0
1	0

· x₁ +

[2^#(x₁)]

0	1
0	0

· x₁ +

[0(x₁)]

0	1
0	0

· x₁ +

[3^#(x₁)]

[2(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

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

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

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

→

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

(153)

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

3^#(2(0(5(2(x1)))))	→	0^#(2(2(2(x1))))	(189)
3^#(2(0(5(2(x1)))))	→	3^#(0(2(2(2(x1)))))	(84)
0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
2^#(0(1(2(x1))))	→	0^#(2(2(1(x1))))	(193)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
2^#(0(1(2(x1))))	→	0^#(x1)	(56)
2^#(0(1(2(x1))))	→	2^#(0(x1))	(127)
2^#(0(1(2(x1))))	→	2^#(2(0(x1)))	(115)
2^#(0(1(2(x1))))	→	3^#(2(2(0(x1))))	(206)
3^#(0(4(2(x1))))	→	0^#(2(x1))	(116)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(3(0(0(5(x1)))))	→	0^#(0(0(x1)))	(149)
3^#(2(0(1(2(x1)))))	→	0^#(2(x1))	(122)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

[4(x₁)]

[5(x₁)]

0	1
0	0

· x₁ +

[3(x₁)]

[2^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

[3^#(x₁)]

1	0
0	0

· x₁ +

[2(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

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

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

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

→

0^#(2(x1))

(122)

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

3^#(2(0(5(2(x1)))))	→	0^#(2(2(2(x1))))	(189)
3^#(2(0(5(2(x1)))))	→	3^#(0(2(2(2(x1)))))	(84)
0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
2^#(0(1(2(x1))))	→	0^#(2(2(1(x1))))	(193)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
2^#(0(1(2(x1))))	→	0^#(x1)	(56)
2^#(0(1(2(x1))))	→	2^#(0(x1))	(127)
2^#(0(1(2(x1))))	→	2^#(2(0(x1)))	(115)
2^#(0(1(2(x1))))	→	3^#(2(2(0(x1))))	(206)
3^#(0(4(2(x1))))	→	0^#(2(x1))	(116)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(3(0(0(5(x1)))))	→	0^#(0(0(x1)))	(149)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	1
0	0

· x₁ +

[4(x₁)]

[5(x₁)]

[3(x₁)]

0	1
0	1

· x₁ +

[2^#(x₁)]

1	0
0	0

· x₁ +

[0(x₁)]

1	0
0	0

· x₁ +

[3^#(x₁)]

[2(x₁)]

0	0
1	1

· x₁ +

together with the usable rules

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

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

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

→

2^#(0(x1))

(127)

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

3^#(2(0(5(2(x1)))))	→	0^#(2(2(2(x1))))	(189)
3^#(2(0(5(2(x1)))))	→	3^#(0(2(2(2(x1)))))	(84)
0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
2^#(0(1(2(x1))))	→	0^#(2(2(1(x1))))	(193)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
2^#(0(1(2(x1))))	→	0^#(x1)	(56)
2^#(0(1(2(x1))))	→	2^#(2(0(x1)))	(115)
2^#(0(1(2(x1))))	→	3^#(2(2(0(x1))))	(206)
3^#(0(4(2(x1))))	→	0^#(2(x1))	(116)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(3(0(0(5(x1)))))	→	0^#(0(0(x1)))	(149)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	1
[1(x₁)]	=	19805
[4(x₁)]	=	1
[5(x₁)]	=	1
[3(x₁)]	=	1
[2^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	19804
[2(x₁)]	=	1

together with the usable rules

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

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

3^#(2(0(5(2(x1)))))	→	0^#(2(2(2(x1))))	(189)
2^#(0(1(2(x1))))	→	0^#(2(2(1(x1))))	(193)
2^#(0(1(2(x1))))	→	0^#(x1)	(56)
2^#(0(1(2(x1))))	→	2^#(2(0(x1)))	(115)
2^#(0(1(2(x1))))	→	3^#(2(2(0(x1))))	(206)
3^#(0(4(2(x1))))	→	0^#(2(x1))	(116)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

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

→

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

(84)

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)]	=	1
[1(x₁)]	=	2247
[4(x₁)]	=	33655
[5(x₁)]	=	12353
[3(x₁)]	=	x₁ + 31408
[2^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	65063
[3^#(x₁)]	=	x₁ + 19804
[2(x₁)]	=	x₁ + 31408

together with the usable rules

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

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

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

→

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

(84)

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 0 components.

The 2^nd component contains the pair

0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(3(0(0(5(x1)))))	→	0^#(0(0(x1)))	(149)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

0	1
0	0

· x₁ +

[1(x₁)]

0	0
0	1

· x₁ +

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

[3(x₁)]

1	0
1	0

· x₁ +

[2^#(x₁)]

1	1
0	0

· x₁ +

[0(x₁)]

0	1
0	0

· x₁ +

[3^#(x₁)]

[2(x₁)]

x₁ +

together with the usable rules

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

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

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

→

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

(149)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

115006

[1(x₁)]

60250

[4(x₁)]

0	1
0	1

· x₁ +

27891

27890

[5(x₁)]

32360

[3(x₁)]

0	1
0	1

· x₁ +

4467

2747

[2^#(x₁)]

1	1
0	0

· x₁ +

[0(x₁)]

1	0
1	0

· x₁ +

[3^#(x₁)]

[2(x₁)]

0	1
0	1

· x₁ +

29611

together with the usable rules

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

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

0^#(5(0(5(x1))))	→	2^#(0(0(5(x1))))	(144)
2^#(0(4(2(x1))))	→	0^#(2(2(1(x1))))	(111)
2^#(0(4(2(x1))))	→	0^#(2(1(2(1(x1)))))	(99)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

115006

[1(x₁)]

0	1
0	0

· x₁ +

[4(x₁)]

1	0
1	1

· x₁ +

[5(x₁)]

1	0
0	0

· x₁ +

[3(x₁)]

0	0
0	1

· x₁ +

[2^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

13999

[3^#(x₁)]

[2(x₁)]

0	0
1	1

· x₁ +

together with the usable rules

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

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

0^#(1(0(4(x1))))	→	0^#(0(x1))	(105)
0^#(1(0(5(x1))))	→	0^#(0(x1))	(168)
0^#(1(2(0(5(x1)))))	→	0^#(5(2(1(0(x1)))))	(152)
0^#(1(0(5(5(x1)))))	→	0^#(0(5(x1)))	(176)
0^#(1(0(5(x1))))	→	0^#(0(2(x1)))	(58)
0^#(1(0(4(x1))))	→	0^#(5(0(x1)))	(64)
0^#(5(1(0(4(x1)))))	→	0^#(0(x1))	(186)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
1	0

· x₁ +

115006

[1(x₁)]

0	1
0	0

· x₁ +

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

x₁ +

[3(x₁)]

0	0
0	1

· x₁ +

[2^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

[3^#(x₁)]

[2(x₁)]

0	0
1	0

· x₁ +

together with the usable rules

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

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

0^#(0(1(2(x1))))	→	0^#(0(x1))	(98)
0^#(0(1(2(2(x1)))))	→	0^#(0(2(1(2(1(x1))))))	(159)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
0^#(1(4(2(x1))))	→	0^#(2(x1))	(85)
0^#(1(4(2(2(x1)))))	→	0^#(2(4(2(x1))))	(121)
0^#(0(1(2(x1))))	→	0^#(0(2(x1)))	(51)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

115006

[1(x₁)]

285215

700

[4(x₁)]

54813

317

[5(x₁)]

1	1
0	0

· x₁ +

[3(x₁)]

0	1
0	1

· x₁ +

54496

61396

[2^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

116527

[3^#(x₁)]

[2(x₁)]

0	0
1	0

· x₁ +

55196

174823

together with the usable rules

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

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

0^#(5(0(5(x1))))	→	0^#(0(5(x1)))	(87)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)
0^#(5(0(5(x1))))	→	0^#(0(x1))	(82)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.