Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/39849)

The rewrite relation of the following TRS is considered.

0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)

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 164 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

2^#(2(0(3(x1))))	→	2^#(0(2(1(3(x1)))))	(150)
0^#(1(2(5(x1))))	→	0^#(5(x1))	(221)
2^#(2(0(3(x1))))	→	0^#(2(1(3(x1))))	(148)
0^#(1(0(3(x1))))	→	3^#(0(x1))	(219)
0^#(1(4(2(x1))))	→	0^#(2(1(4(x1))))	(145)
0^#(5(2(0(x1))))	→	5^#(0(0(x1)))	(143)
3^#(0(1(0(x1))))	→	0^#(0(x1))	(216)
3^#(1(0(3(x1))))	→	3^#(0(x1))	(138)
0^#(1(4(5(x1))))	→	4^#(x1)	(135)
0^#(3(0(2(x1))))	→	3^#(1(x1))	(134)
0^#(3(0(1(x1))))	→	0^#(1(3(1(x1))))	(213)
0^#(2(0(3(x1))))	→	3^#(1(x1))	(212)
3^#(0(1(0(x1))))	→	0^#(0(0(x1)))	(131)
0^#(1(4(2(x1))))	→	0^#(2(1(x1)))	(130)
5^#(1(0(3(x1))))	→	0^#(x1)	(209)
3^#(2(0(1(x1))))	→	0^#(2(1(x1)))	(63)
0^#(1(4(2(x1))))	→	4^#(x1)	(208)
0^#(1(4(5(x1))))	→	5^#(4(x1))	(128)
2^#(5(0(3(x1))))	→	0^#(2(1(3(x1))))	(207)
0^#(2(0(3(x1))))	→	2^#(3(1(x1)))	(206)
0^#(3(0(3(x1))))	→	3^#(1(3(x1)))	(125)
0^#(3(2(x1)))	→	0^#(2(1(3(x1))))	(205)
0^#(1(4(2(x1))))	→	2^#(1(x1))	(123)
5^#(1(0(3(x1))))	→	3^#(0(x1))	(203)
3^#(1(0(3(x1))))	→	0^#(x1)	(122)
0^#(3(2(x1)))	→	3^#(x1)	(202)
4^#(4(1(2(x1))))	→	4^#(x1)	(121)
3^#(5(1(4(x1))))	→	5^#(4(x1))	(201)
2^#(3(1(4(x1))))	→	3^#(4(x1))	(119)
5^#(2(0(5(x1))))	→	2^#(1(5(x1)))	(118)
2^#(2(0(3(x1))))	→	2^#(1(3(x1)))	(198)
0^#(0(3(2(x1))))	→	2^#(1(3(x1)))	(116)
0^#(1(4(2(x1))))	→	4^#(0(2(1(x1))))	(197)
0^#(3(0(1(x1))))	→	3^#(1(x1))	(196)
5^#(1(4(3(x1))))	→	5^#(x1)	(195)
3^#(0(1(0(x1))))	→	3^#(1(0(0(0(x1)))))	(194)
0^#(1(0(3(x1))))	→	0^#(x1)	(192)
5^#(1(4(4(x1))))	→	5^#(4(x1))	(193)
3^#(0(3(0(x1))))	→	0^#(0(x1))	(114)
3^#(0(3(0(x1))))	→	3^#(1(3(0(0(x1)))))	(113)
0^#(5(2(0(x1))))	→	0^#(0(x1))	(190)
0^#(1(4(2(x1))))	→	0^#(0(2(1(4(x1)))))	(111)
2^#(5(0(3(x1))))	→	2^#(1(3(x1)))	(189)
3^#(4(1(2(x1))))	→	4^#(x1)	(187)
0^#(3(2(0(x1))))	→	0^#(2(1(3(0(x1)))))	(108)
0^#(1(4(2(x1))))	→	0^#(4(0(2(1(x1)))))	(106)
0^#(1(2(3(x1))))	→	3^#(1(x1))	(182)
2^#(1(2(3(x1))))	→	2^#(x1)	(183)
2^#(1(2(3(x1))))	→	3^#(2(x1))	(102)
0^#(3(0(2(x1))))	→	0^#(0(2(3(1(x1)))))	(181)
0^#(0(3(2(x1))))	→	0^#(2(1(3(x1))))	(100)
0^#(0(3(2(x1))))	→	0^#(0(2(1(3(x1)))))	(98)
0^#(0(3(2(x1))))	→	3^#(x1)	(176)
0^#(5(2(0(x1))))	→	2^#(1(5(0(0(x1)))))	(95)
5^#(4(1(0(x1))))	→	5^#(0(x1))	(96)
0^#(2(0(3(x1))))	→	0^#(0(2(3(1(x1)))))	(92)
5^#(1(4(3(x1))))	→	3^#(5(x1))	(174)
5^#(2(0(5(x1))))	→	5^#(0(2(1(5(x1)))))	(90)
2^#(5(0(3(x1))))	→	5^#(0(2(1(3(x1)))))	(87)
5^#(2(0(5(x1))))	→	0^#(2(1(5(x1))))	(88)
0^#(2(0(3(x1))))	→	0^#(2(3(1(x1))))	(85)
0^#(3(0(3(x1))))	→	0^#(3(1(3(x1))))	(172)
0^#(1(4(2(x1))))	→	2^#(1(4(x1)))	(82)
0^#(1(2(3(x1))))	→	0^#(2(1(3(1(x1)))))	(169)
4^#(1(4(3(x1))))	→	3^#(4(x1))	(168)
3^#(2(0(1(x1))))	→	2^#(1(x1))	(78)
0^#(3(2(0(x1))))	→	2^#(1(3(0(x1))))	(79)
0^#(3(0(1(x1))))	→	0^#(0(1(3(1(x1)))))	(166)
3^#(2(0(1(x1))))	→	2^#(1(x1))	(78)
0^#(3(0(2(x1))))	→	2^#(3(1(x1)))	(77)
0^#(3(0(2(x1))))	→	0^#(2(3(1(x1))))	(165)
3^#(0(1(4(x1))))	→	0^#(4(x1))	(75)
5^#(4(1(0(x1))))	→	4^#(0(x1))	(160)
0^#(1(2(3(x1))))	→	2^#(1(3(1(x1))))	(71)
3^#(2(0(1(x1))))	→	3^#(0(2(1(x1))))	(72)
0^#(3(2(0(x1))))	→	3^#(0(x1))	(159)
3^#(2(0(1(x1))))	→	3^#(1(0(2(1(x1)))))	(69)
3^#(0(3(0(x1))))	→	3^#(0(0(x1)))	(158)
4^#(1(4(3(x1))))	→	4^#(x1)	(67)
0^#(3(0(3(x1))))	→	0^#(0(3(1(3(x1)))))	(152)
3^#(2(0(1(x1))))	→	0^#(2(1(x1)))	(63)
0^#(3(2(x1)))	→	2^#(1(3(x1)))	(61)

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

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

0^#(1(2(5(x1))))	→	0^#(5(x1))	(221)
2^#(2(0(3(x1))))	→	0^#(2(1(3(x1))))	(148)
0^#(1(0(3(x1))))	→	3^#(0(x1))	(219)
0^#(5(2(0(x1))))	→	5^#(0(0(x1)))	(143)
3^#(0(1(0(x1))))	→	0^#(0(x1))	(216)
3^#(1(0(3(x1))))	→	3^#(0(x1))	(138)
0^#(1(4(5(x1))))	→	4^#(x1)	(135)
0^#(3(0(2(x1))))	→	3^#(1(x1))	(134)
0^#(2(0(3(x1))))	→	3^#(1(x1))	(212)
3^#(0(1(0(x1))))	→	0^#(0(0(x1)))	(131)
0^#(1(4(2(x1))))	→	0^#(2(1(x1)))	(130)
5^#(1(0(3(x1))))	→	0^#(x1)	(209)
3^#(2(0(1(x1))))	→	0^#(2(1(x1)))	(63)
0^#(1(4(2(x1))))	→	4^#(x1)	(208)
0^#(1(4(5(x1))))	→	5^#(4(x1))	(128)
2^#(5(0(3(x1))))	→	0^#(2(1(3(x1))))	(207)
0^#(2(0(3(x1))))	→	2^#(3(1(x1)))	(206)
0^#(3(0(3(x1))))	→	3^#(1(3(x1)))	(125)
0^#(1(4(2(x1))))	→	2^#(1(x1))	(123)
5^#(1(0(3(x1))))	→	3^#(0(x1))	(203)
3^#(1(0(3(x1))))	→	0^#(x1)	(122)
0^#(3(2(x1)))	→	3^#(x1)	(202)
4^#(4(1(2(x1))))	→	4^#(x1)	(121)
3^#(5(1(4(x1))))	→	5^#(4(x1))	(201)
2^#(3(1(4(x1))))	→	3^#(4(x1))	(119)
5^#(2(0(5(x1))))	→	2^#(1(5(x1)))	(118)
2^#(2(0(3(x1))))	→	2^#(1(3(x1)))	(198)
0^#(0(3(2(x1))))	→	2^#(1(3(x1)))	(116)
0^#(1(4(2(x1))))	→	4^#(0(2(1(x1))))	(197)
0^#(3(0(1(x1))))	→	3^#(1(x1))	(196)
5^#(1(4(3(x1))))	→	5^#(x1)	(195)
0^#(1(0(3(x1))))	→	0^#(x1)	(192)
5^#(1(4(4(x1))))	→	5^#(4(x1))	(193)
3^#(0(3(0(x1))))	→	0^#(0(x1))	(114)
0^#(5(2(0(x1))))	→	0^#(0(x1))	(190)
2^#(5(0(3(x1))))	→	2^#(1(3(x1)))	(189)
3^#(4(1(2(x1))))	→	4^#(x1)	(187)
0^#(1(2(3(x1))))	→	3^#(1(x1))	(182)
2^#(1(2(3(x1))))	→	2^#(x1)	(183)
2^#(1(2(3(x1))))	→	3^#(2(x1))	(102)
0^#(0(3(2(x1))))	→	3^#(x1)	(176)
0^#(5(2(0(x1))))	→	2^#(1(5(0(0(x1)))))	(95)
5^#(4(1(0(x1))))	→	5^#(0(x1))	(96)
5^#(1(4(3(x1))))	→	3^#(5(x1))	(174)
2^#(5(0(3(x1))))	→	5^#(0(2(1(3(x1)))))	(87)
5^#(2(0(5(x1))))	→	0^#(2(1(5(x1))))	(88)
0^#(1(4(2(x1))))	→	2^#(1(4(x1)))	(82)
4^#(1(4(3(x1))))	→	3^#(4(x1))	(168)
3^#(2(0(1(x1))))	→	2^#(1(x1))	(78)
0^#(3(2(0(x1))))	→	2^#(1(3(0(x1))))	(79)
3^#(2(0(1(x1))))	→	2^#(1(x1))	(78)
0^#(3(0(2(x1))))	→	2^#(3(1(x1)))	(77)
3^#(0(1(4(x1))))	→	0^#(4(x1))	(75)
5^#(4(1(0(x1))))	→	4^#(0(x1))	(160)
0^#(1(2(3(x1))))	→	2^#(1(3(1(x1))))	(71)
0^#(3(2(0(x1))))	→	3^#(0(x1))	(159)
3^#(0(3(0(x1))))	→	3^#(0(0(x1)))	(158)
4^#(1(4(3(x1))))	→	4^#(x1)	(67)
3^#(2(0(1(x1))))	→	0^#(2(1(x1)))	(63)
0^#(3(2(x1)))	→	2^#(1(3(x1)))	(61)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

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

→

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

(150)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

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

→

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

(150)

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

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

→

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

(72)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	1
0	0

· x₁ +

[4(x₁)]

1	0
1	0

· x₁ +

[5(x₁)]

0	0
1	0

· x₁ +

[3(x₁)]

[2^#(x₁)]

[4^#(x₁)]

[0(x₁)]

[3^#(x₁)]

0	1
1	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

0	1
0	1

· x₁ +

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

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

→

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

(72)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

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

→

5^#(0(2(1(5(x1)))))

(90)

1.1.1.1.3 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)]

1	0
0	0

· x₁ +

[5(x₁)]

[3(x₁)]

1	1
0	0

· x₁ +

19973

[2^#(x₁)]

[4^#(x₁)]

[0(x₁)]

0	0
1	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

0	1
0	0

· x₁ +

[2(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

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

→

5^#(0(2(1(5(x1)))))

(90)

could be deleted.

1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

0^#(3(0(2(x1))))	→	0^#(2(3(1(x1))))	(165)
0^#(3(0(2(x1))))	→	0^#(0(2(3(1(x1)))))	(181)
0^#(0(3(2(x1))))	→	0^#(2(1(3(x1))))	(100)
0^#(0(3(2(x1))))	→	0^#(0(2(1(3(x1)))))	(98)
0^#(1(2(3(x1))))	→	0^#(2(1(3(1(x1)))))	(169)
0^#(2(0(3(x1))))	→	0^#(2(3(1(x1))))	(85)
0^#(2(0(3(x1))))	→	0^#(0(2(3(1(x1)))))	(92)
0^#(3(0(3(x1))))	→	0^#(3(1(3(x1))))	(172)
0^#(3(0(3(x1))))	→	0^#(0(3(1(3(x1)))))	(152)
0^#(3(0(1(x1))))	→	0^#(1(3(1(x1))))	(213)
0^#(3(0(1(x1))))	→	0^#(0(1(3(1(x1)))))	(166)
0^#(3(2(0(x1))))	→	0^#(2(1(3(0(x1)))))	(108)
0^#(1(4(2(x1))))	→	0^#(4(0(2(1(x1)))))	(106)
0^#(1(4(2(x1))))	→	0^#(2(1(4(x1))))	(145)
0^#(1(4(2(x1))))	→	0^#(0(2(1(4(x1)))))	(111)
0^#(3(2(x1)))	→	0^#(2(1(3(x1))))	(205)

1.1.1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	59858
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	59858
[2^#(x₁)]	=	2
[4^#(x₁)]	=	0
[0(x₁)]	=	31082
[3^#(x₁)]	=	2
[5^#(x₁)]	=	2
[2(x₁)]	=	31082

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

0^#(3(0(2(x1))))	→	0^#(2(3(1(x1))))	(165)
0^#(3(0(2(x1))))	→	0^#(0(2(3(1(x1)))))	(181)
0^#(1(2(3(x1))))	→	0^#(2(1(3(1(x1)))))	(169)
0^#(3(0(3(x1))))	→	0^#(0(3(1(3(x1)))))	(152)
0^#(3(0(1(x1))))	→	0^#(0(1(3(1(x1)))))	(166)
0^#(3(2(0(x1))))	→	0^#(2(1(3(0(x1)))))	(108)
0^#(1(4(2(x1))))	→	0^#(4(0(2(1(x1)))))	(106)
0^#(1(4(2(x1))))	→	0^#(2(1(4(x1))))	(145)
0^#(1(4(2(x1))))	→	0^#(0(2(1(4(x1)))))	(111)
0^#(3(2(x1)))	→	0^#(2(1(3(x1))))	(205)

could be deleted.

1.1.1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(0(3(2(x1))))	→	0^#(2(1(3(x1))))	(100)
0^#(0(3(2(x1))))	→	0^#(0(2(1(3(x1)))))	(98)
0^#(2(0(3(x1))))	→	0^#(2(3(1(x1))))	(85)
0^#(2(0(3(x1))))	→	0^#(0(2(3(1(x1)))))	(92)
0^#(3(0(3(x1))))	→	0^#(3(1(3(x1))))	(172)

1.1.1.1.4.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	1
[4(x₁)]	=	1
[5(x₁)]	=	1
[3(x₁)]	=	x₁ + 32112
[2^#(x₁)]	=	2
[4^#(x₁)]	=	0
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	2
[5^#(x₁)]	=	2
[2(x₁)]	=	1

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

0^#(0(3(2(x1))))	→	0^#(2(1(3(x1))))	(100)
0^#(0(3(2(x1))))	→	0^#(0(2(1(3(x1)))))	(98)
0^#(3(0(3(x1))))	→	0^#(3(1(3(x1))))	(172)

could be deleted.

1.1.1.1.4.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(2(0(3(x1))))	→	0^#(2(3(1(x1))))	(85)
0^#(2(0(3(x1))))	→	0^#(0(2(3(1(x1)))))	(92)

1.1.1.1.4.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

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

→

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

(85)

could be deleted.

1.1.1.1.4.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

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

(92)

1.1.1.1.4.1.1.1.1.1.1 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	0

· x₁ +

[1(x₁)]

[4(x₁)]

1	0
0	0

· x₁ +

[5(x₁)]

[3(x₁)]

1	1
0	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

[0(x₁)]

0	0
1	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

0	1
0	0

· x₁ +

[2(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

0(3(0(2(x1))))	→	0(0(2(3(1(x1)))))	(18)
5(1(0(3(x1))))	→	5(1(1(3(0(x1)))))	(50)
0(0(3(2(x1))))	→	0(0(2(1(3(x1)))))	(4)
0(1(5(2(x1))))	→	0(2(5(1(1(x1)))))	(15)
0(1(2(3(x1))))	→	0(2(1(3(1(x1)))))	(8)
5(1(5(3(x1))))	→	5(5(1(1(3(x1)))))	(54)
0(1(2(x1)))	→	0(2(1(1(x1))))	(1)
2(4(1(x1)))	→	4(2(1(1(x1))))	(3)
0(2(0(3(x1))))	→	0(0(2(3(1(x1)))))	(16)
0(3(5(1(x1))))	→	0(5(3(1(1(x1)))))	(21)
3(1(0(3(x1))))	→	3(1(1(3(0(x1)))))	(36)
2(2(0(3(x1))))	→	2(0(2(1(3(x1)))))	(26)
0(3(0(3(x1))))	→	0(0(3(1(3(x1)))))	(19)
2(5(0(3(x1))))	→	5(0(2(1(3(x1)))))	(32)
0(3(0(1(x1))))	→	0(0(1(3(1(x1)))))	(17)
5(4(1(0(x1))))	→	5(1(1(4(0(x1)))))	(60)
2(2(4(1(x1))))	→	2(4(2(1(1(x1)))))	(27)
3(0(1(4(x1))))	→	3(1(1(0(4(x1)))))	(34)
0(4(0(1(x1))))	→	0(0(4(1(1(x1)))))	(22)
2(3(1(4(x1))))	→	2(1(1(3(4(x1)))))	(28)
4(1(2(1(x1))))	→	2(4(1(1(1(x1)))))	(44)
0(1(0(3(x1))))	→	0(1(1(3(0(x1)))))	(5)
3(0(1(0(x1))))	→	3(1(0(0(0(x1)))))	(33)
0(1(4(1(x1))))	→	0(4(1(1(1(x1)))))	(10)
3(2(0(1(x1))))	→	1(3(0(2(1(x1)))))	(39)
0(1(2(2(x1))))	→	0(2(1(1(2(x1)))))	(7)
0(3(2(0(x1))))	→	0(2(1(3(0(x1)))))	(20)
2(1(4(1(x1))))	→	4(2(1(1(1(x1)))))	(25)
5(1(0(3(x1))))	→	5(0(1(1(3(x1)))))	(49)
5(1(4(3(x1))))	→	4(1(1(3(5(x1)))))	(52)
2(4(3(1(x1))))	→	4(2(3(1(1(x1)))))	(30)
0(1(4(5(x1))))	→	0(1(1(5(4(x1)))))	(14)
5(2(0(5(x1))))	→	5(0(2(1(5(x1)))))	(56)
2(5(0(1(x1))))	→	0(2(5(1(1(x1)))))	(31)
0(1(4(2(x1))))	→	0(4(0(2(1(x1)))))	(12)
4(1(4(3(x1))))	→	4(1(1(3(4(x1)))))	(45)
0(5(2(0(x1))))	→	2(1(5(0(0(x1)))))	(23)
2(1(2(3(x1))))	→	2(1(1(3(2(x1)))))	(24)
5(3(4(1(x1))))	→	5(4(3(1(1(x1)))))	(57)
0(1(4(2(x1))))	→	0(0(2(1(4(x1)))))	(11)
0(1(2(5(x1))))	→	2(1(1(0(5(x1)))))	(9)
0(1(4(3(x1))))	→	4(0(1(1(3(x1)))))	(13)
5(1(3(1(x1))))	→	5(3(1(1(1(x1)))))	(51)
3(2(0(1(x1))))	→	3(1(0(2(1(x1)))))	(40)
5(2(0(1(x1))))	→	2(0(5(1(1(x1)))))	(55)
5(4(1(0(x1))))	→	4(1(1(5(0(x1)))))	(59)
0(1(0(5(x1))))	→	0(0(1(1(5(x1)))))	(6)
3(1(2(1(x1))))	→	3(2(1(1(1(x1)))))	(38)
5(4(0(1(x1))))	→	5(0(4(1(1(x1)))))	(58)
4(5(0(1(x1))))	→	5(0(4(1(1(x1)))))	(48)
5(1(4(4(x1))))	→	4(1(1(5(4(x1)))))	(53)
4(4(1(2(x1))))	→	4(2(1(1(4(x1)))))	(47)
3(1(2(1(x1))))	→	1(3(2(1(1(x1)))))	(37)
3(4(1(2(x1))))	→	3(2(1(1(4(x1)))))	(41)
3(5(1(4(x1))))	→	3(1(1(5(4(x1)))))	(42)
4(2(4(1(x1))))	→	4(4(2(1(1(x1)))))	(46)
3(0(3(0(x1))))	→	3(1(3(0(0(x1)))))	(35)
2(4(1(4(x1))))	→	4(4(2(1(1(x1)))))	(29)
4(1(0(1(x1))))	→	4(0(1(1(1(x1)))))	(43)
0(3(2(x1)))	→	0(2(1(3(x1))))	(2)

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

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

→

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

(92)

could be deleted.

1.1.1.1.4.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.