Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212308)

The rewrite relation of the following TRS is considered.

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

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 158 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^#(2(5(4(x1))))	→	2^#(4(5(0(x1))))	(136)
3^#(0(5(0(x1))))	→	0^#(3(0(1(1(x1)))))	(135)
4^#(4(5(0(0(x1)))))	→	4^#(0(1(5(0(x1)))))	(200)
0^#(2(5(4(x1))))	→	4^#(5(0(x1)))	(134)
4^#(2(3(5(4(x1)))))	→	2^#(4(4(5(3(x1)))))	(133)
0^#(0(0(4(x1))))	→	2^#(0(2(0(0(4(x1))))))	(131)
5^#(0(3(x1)))	→	3^#(0(1(1(5(x1)))))	(132)
4^#(0(1(0(3(x1)))))	→	0^#(3(0(1(1(x1)))))	(129)
4^#(2(5(0(4(x1)))))	→	4^#(4(1(0(5(2(x1))))))	(130)
0^#(5(5(4(x1))))	→	2^#(4(5(5(1(0(x1))))))	(126)
4^#(1(4(4(0(x1)))))	→	4^#(x1)	(127)
3^#(0(x1))	→	3^#(x1)	(195)
4^#(4(5(0(0(x1)))))	→	5^#(0(x1))	(125)
3^#(0(2(0(x1))))	→	0^#(3(0(1(2(1(x1))))))	(124)
0^#(2(5(4(x1))))	→	2^#(0(x1))	(123)
4^#(4(5(0(0(x1)))))	→	4^#(4(0(1(5(0(x1))))))	(121)
4^#(2(5(0(4(x1)))))	→	4^#(1(0(5(2(x1)))))	(120)
3^#(0(4(0(x1))))	→	4^#(x1)	(193)
0^#(4(1(0(x1))))	→	4^#(x1)	(192)
3^#(2(1(4(x1))))	→	2^#(4(3(x1)))	(117)
5^#(1(0(3(x1))))	→	5^#(3(0(1(1(1(x1))))))	(190)
0^#(0(2(4(x1))))	→	0^#(2(2(0(4(x1)))))	(115)
0^#(0(0(4(x1))))	→	2^#(0(0(4(x1))))	(189)
3^#(1(0(4(x1))))	→	0^#(x1)	(114)
0^#(2(3(0(x1))))	→	4^#(0(0(x1)))	(188)
3^#(0(4(0(x1))))	→	3^#(0(0(1(1(4(x1))))))	(113)
0^#(2(5(4(x1))))	→	2^#(2(4(5(0(x1)))))	(111)
5^#(5(0(4(x1))))	→	5^#(4(x1))	(110)
3^#(1(0(4(x1))))	→	4^#(0(x1))	(185)
3^#(2(1(4(x1))))	→	4^#(3(x1))	(109)
3^#(0(2(0(x1))))	→	3^#(0(1(2(1(x1)))))	(108)
4^#(2(3(5(4(x1)))))	→	2^#(2(4(4(5(3(x1))))))	(107)
3^#(0(5(0(x1))))	→	3^#(0(1(1(x1))))	(105)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
5^#(5(4(3(x1))))	→	4^#(5(5(1(x1))))	(181)
0^#(2(5(4(x1))))	→	5^#(0(x1))	(180)
2^#(3(0(0(x1))))	→	3^#(0(x1))	(103)
3^#(2(1(4(x1))))	→	2^#(2(4(3(x1))))	(179)
0^#(2(3(0(x1))))	→	2^#(3(4(0(0(x1)))))	(178)
3^#(0(0(3(x1))))	→	3^#(0(1(1(x1))))	(177)
4^#(1(0(0(x1))))	→	4^#(1(x1))	(176)
4^#(2(3(5(4(x1)))))	→	4^#(4(5(3(x1))))	(175)
5^#(1(0(3(x1))))	→	3^#(0(1(1(1(x1)))))	(99)
4^#(0(1(0(3(x1)))))	→	3^#(0(1(1(x1))))	(172)
3^#(0(5(0(x1))))	→	5^#(0(3(0(1(1(x1))))))	(96)
5^#(0(3(x1)))	→	5^#(x1)	(97)
0^#(2(3(0(x1))))	→	0^#(0(x1))	(171)
4^#(2(5(0(4(x1)))))	→	0^#(5(2(x1)))	(168)
0^#(2(5(4(x1))))	→	0^#(x1)	(91)
0^#(2(5(4(x1))))	→	0^#(x1)	(91)
2^#(3(0(0(x1))))	→	3^#(0(x1))	(103)
2^#(3(0(0(x1))))	→	2^#(3(0(x1)))	(89)
2^#(5(1(0(4(x1)))))	→	2^#(x1)	(88)
5^#(5(0(4(x1))))	→	5^#(5(1(0(4(x1)))))	(162)
0^#(2(1(3(x1))))	→	4^#(x1)	(76)
4^#(2(3(5(4(x1)))))	→	4^#(5(3(x1)))	(161)
0^#(2(1(3(x1))))	→	2^#(0(1(3(4(x1)))))	(85)
0^#(2(3(0(x1))))	→	2^#(0(1(2(3(0(x1))))))	(160)
4^#(0(1(0(3(x1)))))	→	4^#(0(3(0(1(1(x1))))))	(83)
5^#(5(4(3(x1))))	→	5^#(1(x1))	(84)
0^#(2(5(4(x1))))	→	4^#(5(1(2(0(x1)))))	(159)
5^#(5(0(4(x1))))	→	5^#(1(0(4(x1))))	(80)
0^#(5(5(4(x1))))	→	5^#(5(1(0(x1))))	(81)
3^#(0(0(3(x1))))	→	0^#(3(0(1(1(x1)))))	(158)
0^#(5(5(4(x1))))	→	4^#(5(5(1(0(x1)))))	(77)
0^#(2(1(3(x1))))	→	3^#(4(x1))	(78)
4^#(2(5(0(4(x1)))))	→	2^#(x1)	(157)
0^#(2(3(0(x1))))	→	3^#(4(0(0(x1))))	(75)
0^#(2(1(3(x1))))	→	4^#(x1)	(76)
0^#(0(2(4(x1))))	→	0^#(4(x1))	(74)
3^#(2(1(4(x1))))	→	3^#(1(4(x1)))	(156)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
5^#(5(0(4(x1))))	→	5^#(5(4(x1)))	(155)
0^#(0(0(4(x1))))	→	0^#(2(0(0(4(x1)))))	(70)
4^#(2(3(5(4(x1)))))	→	5^#(3(x1))	(154)
0^#(2(1(3(x1))))	→	2^#(3(4(x1)))	(68)
0^#(0(2(4(x1))))	→	2^#(0(4(x1)))	(69)
3^#(0(0(x1)))	→	3^#(0(1(0(1(1(x1))))))	(67)
2^#(5(1(0(4(x1)))))	→	4^#(2(x1))	(151)
0^#(5(5(4(x1))))	→	5^#(1(0(x1)))	(64)
5^#(5(4(3(x1))))	→	5^#(5(1(x1)))	(63)
4^#(2(5(0(4(x1)))))	→	5^#(2(x1))	(148)
2^#(3(0(x1)))	→	3^#(x1)	(62)
0^#(5(5(4(x1))))	→	0^#(x1)	(59)
0^#(0(2(4(x1))))	→	2^#(2(0(4(x1))))	(57)
0^#(2(1(3(x1))))	→	3^#(4(x1))	(78)
3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
0^#(2(5(4(x1))))	→	5^#(1(2(0(x1))))	(139)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)
5^#(5(4(3(x1))))	→	3^#(4(5(5(1(x1)))))	(54)
4^#(2(3(5(4(x1)))))	→	3^#(x1)	(53)
0^#(2(5(2(3(x1)))))	→	2^#(x1)	(137)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 64559
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 2447
[3(x₁)]	=	x₁ + 9726
[2^#(x₁)]	=	x₁ + 0
[4^#(x₁)]	=	x₁ + 32279
[0(x₁)]	=	x₁ + 32280
[3^#(x₁)]	=	x₁ + 42005
[5^#(x₁)]	=	x₁ + 34726
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

0^#(2(5(4(x1))))	→	2^#(4(5(0(x1))))	(136)
3^#(0(5(0(x1))))	→	0^#(3(0(1(1(x1)))))	(135)
4^#(2(3(5(4(x1)))))	→	2^#(4(4(5(3(x1)))))	(133)
0^#(0(0(4(x1))))	→	2^#(0(2(0(0(4(x1))))))	(131)
0^#(5(5(4(x1))))	→	2^#(4(5(5(1(0(x1))))))	(126)
4^#(1(4(4(0(x1)))))	→	4^#(x1)	(127)
3^#(0(x1))	→	3^#(x1)	(195)
4^#(4(5(0(0(x1)))))	→	5^#(0(x1))	(125)
0^#(2(5(4(x1))))	→	2^#(0(x1))	(123)
3^#(0(4(0(x1))))	→	4^#(x1)	(193)
0^#(4(1(0(x1))))	→	4^#(x1)	(192)
3^#(2(1(4(x1))))	→	2^#(4(3(x1)))	(117)
0^#(0(0(4(x1))))	→	2^#(0(0(4(x1))))	(189)
3^#(1(0(4(x1))))	→	0^#(x1)	(114)
0^#(2(3(0(x1))))	→	4^#(0(0(x1)))	(188)
0^#(2(5(4(x1))))	→	2^#(2(4(5(0(x1)))))	(111)
5^#(5(0(4(x1))))	→	5^#(4(x1))	(110)
3^#(1(0(4(x1))))	→	4^#(0(x1))	(185)
3^#(0(2(0(x1))))	→	3^#(0(1(2(1(x1)))))	(108)
4^#(2(3(5(4(x1)))))	→	2^#(2(4(4(5(3(x1))))))	(107)
3^#(0(5(0(x1))))	→	3^#(0(1(1(x1))))	(105)
5^#(5(4(3(x1))))	→	4^#(5(5(1(x1))))	(181)
2^#(3(0(0(x1))))	→	3^#(0(x1))	(103)
3^#(2(1(4(x1))))	→	2^#(2(4(3(x1))))	(179)
0^#(2(3(0(x1))))	→	2^#(3(4(0(0(x1)))))	(178)
3^#(0(0(3(x1))))	→	3^#(0(1(1(x1))))	(177)
4^#(1(0(0(x1))))	→	4^#(1(x1))	(176)
5^#(1(0(3(x1))))	→	3^#(0(1(1(1(x1)))))	(99)
4^#(0(1(0(3(x1)))))	→	3^#(0(1(1(x1))))	(172)
5^#(0(3(x1)))	→	5^#(x1)	(97)
0^#(2(3(0(x1))))	→	0^#(0(x1))	(171)
0^#(2(5(4(x1))))	→	0^#(x1)	(91)
0^#(2(5(4(x1))))	→	0^#(x1)	(91)
2^#(3(0(0(x1))))	→	3^#(0(x1))	(103)
2^#(3(0(0(x1))))	→	2^#(3(0(x1)))	(89)
2^#(5(1(0(4(x1)))))	→	2^#(x1)	(88)
0^#(2(1(3(x1))))	→	4^#(x1)	(76)
0^#(2(1(3(x1))))	→	2^#(0(1(3(4(x1)))))	(85)
0^#(2(3(0(x1))))	→	2^#(0(1(2(3(0(x1))))))	(160)
5^#(5(4(3(x1))))	→	5^#(1(x1))	(84)
5^#(5(0(4(x1))))	→	5^#(1(0(4(x1))))	(80)
3^#(0(0(3(x1))))	→	0^#(3(0(1(1(x1)))))	(158)
0^#(2(1(3(x1))))	→	3^#(4(x1))	(78)
4^#(2(5(0(4(x1)))))	→	2^#(x1)	(157)
0^#(2(1(3(x1))))	→	4^#(x1)	(76)
0^#(0(2(4(x1))))	→	0^#(4(x1))	(74)
5^#(5(0(4(x1))))	→	5^#(5(4(x1)))	(155)
0^#(2(1(3(x1))))	→	2^#(3(4(x1)))	(68)
0^#(0(2(4(x1))))	→	2^#(0(4(x1)))	(69)
2^#(5(1(0(4(x1)))))	→	4^#(2(x1))	(151)
0^#(5(5(4(x1))))	→	5^#(1(0(x1)))	(64)
5^#(5(4(3(x1))))	→	5^#(5(1(x1)))	(63)
4^#(2(5(0(4(x1)))))	→	5^#(2(x1))	(148)
2^#(3(0(x1)))	→	3^#(x1)	(62)
0^#(5(5(4(x1))))	→	0^#(x1)	(59)
0^#(0(2(4(x1))))	→	2^#(2(0(4(x1))))	(57)
0^#(2(1(3(x1))))	→	3^#(4(x1))	(78)
4^#(2(3(5(4(x1)))))	→	3^#(x1)	(53)
0^#(2(5(2(3(x1)))))	→	2^#(x1)	(137)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(5(0(4(x1))))	→	5^#(5(1(0(4(x1)))))	(162)
3^#(0(2(0(x1))))	→	0^#(3(0(1(2(1(x1))))))	(124)
0^#(2(5(4(x1))))	→	5^#(0(x1))	(180)
0^#(2(5(4(x1))))	→	4^#(5(0(x1)))	(134)
3^#(2(1(4(x1))))	→	3^#(1(4(x1)))	(156)
0^#(2(3(0(x1))))	→	3^#(4(0(0(x1))))	(75)
3^#(0(4(0(x1))))	→	3^#(0(0(1(1(4(x1))))))	(113)
0^#(5(5(4(x1))))	→	5^#(5(1(0(x1))))	(81)
0^#(5(5(4(x1))))	→	4^#(5(5(1(0(x1)))))	(77)
3^#(0(5(0(x1))))	→	5^#(0(3(0(1(1(x1))))))	(96)
0^#(2(5(4(x1))))	→	5^#(1(2(0(x1))))	(139)
0^#(2(5(4(x1))))	→	4^#(5(1(2(0(x1)))))	(159)
3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
4^#(4(5(0(0(x1)))))	→	4^#(0(1(5(0(x1)))))	(200)
4^#(4(5(0(0(x1)))))	→	4^#(4(0(1(5(0(x1))))))	(121)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
3^#(2(1(4(x1))))	→	4^#(3(x1))	(109)
4^#(0(1(0(3(x1)))))	→	0^#(3(0(1(1(x1)))))	(129)
4^#(0(1(0(3(x1)))))	→	4^#(0(3(0(1(1(x1))))))	(83)
0^#(0(0(4(x1))))	→	0^#(2(0(0(4(x1)))))	(70)
0^#(0(2(4(x1))))	→	0^#(2(2(0(4(x1)))))	(115)
5^#(5(4(3(x1))))	→	3^#(4(5(5(1(x1)))))	(54)
4^#(2(5(0(4(x1)))))	→	0^#(5(2(x1)))	(168)
4^#(2(5(0(4(x1)))))	→	4^#(4(1(0(5(2(x1))))))	(130)
4^#(2(3(5(4(x1)))))	→	5^#(3(x1))	(154)
4^#(2(3(5(4(x1)))))	→	4^#(5(3(x1)))	(161)
4^#(2(3(5(4(x1)))))	→	4^#(4(5(3(x1))))	(175)
5^#(1(0(3(x1))))	→	5^#(3(0(1(1(1(x1))))))	(190)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 32279
[1(x₁)]	=	0
[4(x₁)]	=	1
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	0
[2^#(x₁)]	=	0
[4^#(x₁)]	=	32280
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	32280
[5^#(x₁)]	=	32280
[2(x₁)]	=	1

together with the usable rules

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

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

3^#(0(2(0(x1))))	→	0^#(3(0(1(2(1(x1))))))	(124)
4^#(0(1(0(3(x1)))))	→	0^#(3(0(1(1(x1)))))	(129)

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

5^#(5(0(4(x1))))	→	5^#(5(1(0(4(x1)))))	(162)
0^#(2(5(4(x1))))	→	5^#(0(x1))	(180)
0^#(2(5(4(x1))))	→	4^#(5(0(x1)))	(134)
3^#(2(1(4(x1))))	→	3^#(1(4(x1)))	(156)
0^#(2(3(0(x1))))	→	3^#(4(0(0(x1))))	(75)
3^#(0(4(0(x1))))	→	3^#(0(0(1(1(4(x1))))))	(113)
0^#(5(5(4(x1))))	→	5^#(5(1(0(x1))))	(81)
0^#(5(5(4(x1))))	→	4^#(5(5(1(0(x1)))))	(77)
3^#(0(5(0(x1))))	→	5^#(0(3(0(1(1(x1))))))	(96)
0^#(2(5(4(x1))))	→	5^#(1(2(0(x1))))	(139)
0^#(2(5(4(x1))))	→	4^#(5(1(2(0(x1)))))	(159)
3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
4^#(4(5(0(0(x1)))))	→	4^#(0(1(5(0(x1)))))	(200)
4^#(4(5(0(0(x1)))))	→	4^#(4(0(1(5(0(x1))))))	(121)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
3^#(2(1(4(x1))))	→	4^#(3(x1))	(109)
4^#(0(1(0(3(x1)))))	→	4^#(0(3(0(1(1(x1))))))	(83)
0^#(0(0(4(x1))))	→	0^#(2(0(0(4(x1)))))	(70)
0^#(0(2(4(x1))))	→	0^#(2(2(0(4(x1)))))	(115)
5^#(5(4(3(x1))))	→	3^#(4(5(5(1(x1)))))	(54)
4^#(2(5(0(4(x1)))))	→	0^#(5(2(x1)))	(168)
4^#(2(5(0(4(x1)))))	→	4^#(4(1(0(5(2(x1))))))	(130)
4^#(2(3(5(4(x1)))))	→	5^#(3(x1))	(154)
4^#(2(3(5(4(x1)))))	→	4^#(5(3(x1)))	(161)
4^#(2(3(5(4(x1)))))	→	4^#(4(5(3(x1))))	(175)
5^#(1(0(3(x1))))	→	5^#(3(0(1(1(1(x1))))))	(190)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 24773
[1(x₁)]	=	0
[4(x₁)]	=	7507
[5(x₁)]	=	7507
[3(x₁)]	=	x₁ + 0
[2^#(x₁)]	=	0
[4^#(x₁)]	=	32280
[0(x₁)]	=	x₁ + 1
[3^#(x₁)]	=	32280
[5^#(x₁)]	=	32280
[2(x₁)]	=	7507

together with the usable rules

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

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

0^#(0(0(4(x1))))	→	0^#(2(0(0(4(x1)))))	(70)
0^#(0(2(4(x1))))	→	0^#(2(2(0(4(x1)))))	(115)

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

5^#(5(0(4(x1))))	→	5^#(5(1(0(4(x1)))))	(162)
0^#(2(5(4(x1))))	→	5^#(0(x1))	(180)
0^#(2(5(4(x1))))	→	4^#(5(0(x1)))	(134)
3^#(2(1(4(x1))))	→	3^#(1(4(x1)))	(156)
0^#(2(3(0(x1))))	→	3^#(4(0(0(x1))))	(75)
3^#(0(4(0(x1))))	→	3^#(0(0(1(1(4(x1))))))	(113)
0^#(5(5(4(x1))))	→	5^#(5(1(0(x1))))	(81)
0^#(5(5(4(x1))))	→	4^#(5(5(1(0(x1)))))	(77)
3^#(0(5(0(x1))))	→	5^#(0(3(0(1(1(x1))))))	(96)
0^#(2(5(4(x1))))	→	5^#(1(2(0(x1))))	(139)
0^#(2(5(4(x1))))	→	4^#(5(1(2(0(x1)))))	(159)
3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
4^#(4(5(0(0(x1)))))	→	4^#(0(1(5(0(x1)))))	(200)
4^#(4(5(0(0(x1)))))	→	4^#(4(0(1(5(0(x1))))))	(121)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
3^#(2(1(4(x1))))	→	4^#(3(x1))	(109)
4^#(0(1(0(3(x1)))))	→	4^#(0(3(0(1(1(x1))))))	(83)
5^#(5(4(3(x1))))	→	3^#(4(5(5(1(x1)))))	(54)
4^#(2(5(0(4(x1)))))	→	0^#(5(2(x1)))	(168)
4^#(2(5(0(4(x1)))))	→	4^#(4(1(0(5(2(x1))))))	(130)
4^#(2(3(5(4(x1)))))	→	5^#(3(x1))	(154)
4^#(2(3(5(4(x1)))))	→	4^#(5(3(x1)))	(161)
4^#(2(3(5(4(x1)))))	→	4^#(4(5(3(x1))))	(175)
5^#(1(0(3(x1))))	→	5^#(3(0(1(1(1(x1))))))	(190)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(0^#)

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

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

and the following Max-polynomial interpretation

[1(x₁)]	=	0
[4(x₁)]	=	24367
[5(x₁)]	=	24367
[3(x₁)]	=	x₁ + 0
[2^#(x₁)]	=	1
[4^#(x₁)]	=	24367
[0(x₁)]	=	x₁ + 1
[3^#(x₁)]	=	24367
[5^#(x₁)]	=	24367
[2(x₁)]	=	24367

together with the usable rules

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

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

0^#(2(5(4(x1))))	→	5^#(0(x1))	(180)
0^#(2(5(4(x1))))	→	4^#(5(0(x1)))	(134)
0^#(2(3(0(x1))))	→	3^#(4(0(0(x1))))	(75)
0^#(2(5(4(x1))))	→	5^#(1(2(0(x1))))	(139)
0^#(2(5(4(x1))))	→	4^#(5(1(2(0(x1)))))	(159)

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

5^#(5(0(4(x1))))	→	5^#(5(1(0(4(x1)))))	(162)
3^#(2(1(4(x1))))	→	3^#(1(4(x1)))	(156)
3^#(0(4(0(x1))))	→	3^#(0(0(1(1(4(x1))))))	(113)
0^#(5(5(4(x1))))	→	5^#(5(1(0(x1))))	(81)
0^#(5(5(4(x1))))	→	4^#(5(5(1(0(x1)))))	(77)
3^#(0(5(0(x1))))	→	5^#(0(3(0(1(1(x1))))))	(96)
3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
4^#(4(5(0(0(x1)))))	→	4^#(0(1(5(0(x1)))))	(200)
4^#(4(5(0(0(x1)))))	→	4^#(4(0(1(5(0(x1))))))	(121)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
3^#(2(1(4(x1))))	→	4^#(3(x1))	(109)
4^#(0(1(0(3(x1)))))	→	4^#(0(3(0(1(1(x1))))))	(83)
5^#(5(4(3(x1))))	→	3^#(4(5(5(1(x1)))))	(54)
4^#(2(5(0(4(x1)))))	→	0^#(5(2(x1)))	(168)
4^#(2(5(0(4(x1)))))	→	4^#(4(1(0(5(2(x1))))))	(130)
4^#(2(3(5(4(x1)))))	→	5^#(3(x1))	(154)
4^#(2(3(5(4(x1)))))	→	4^#(5(3(x1)))	(161)
4^#(2(3(5(4(x1)))))	→	4^#(4(5(3(x1))))	(175)
5^#(1(0(3(x1))))	→	5^#(3(0(1(1(1(x1))))))	(190)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 32278
[1(x₁)]	=	0
[4(x₁)]	=	1
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	0
[2^#(x₁)]	=	0
[4^#(x₁)]	=	x₁ + 32279
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	32279
[5^#(x₁)]	=	32279
[2(x₁)]	=	1

together with the usable rules

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

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

4^#(4(5(0(0(x1)))))	→	4^#(0(1(5(0(x1)))))	(200)
4^#(2(5(0(4(x1)))))	→	0^#(5(2(x1)))	(168)
4^#(2(3(5(4(x1)))))	→	5^#(3(x1))	(154)
4^#(2(3(5(4(x1)))))	→	4^#(5(3(x1)))	(161)

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

5^#(5(0(4(x1))))	→	5^#(5(1(0(4(x1)))))	(162)
3^#(2(1(4(x1))))	→	3^#(1(4(x1)))	(156)
3^#(0(4(0(x1))))	→	3^#(0(0(1(1(4(x1))))))	(113)
3^#(0(5(0(x1))))	→	5^#(0(3(0(1(1(x1))))))	(96)
3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
5^#(5(4(3(x1))))	→	3^#(4(5(5(1(x1)))))	(54)
5^#(1(0(3(x1))))	→	5^#(3(0(1(1(1(x1))))))	(190)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)

1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 32278
[1(x₁)]	=	0
[4(x₁)]	=	23442
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	0
[2^#(x₁)]	=	0
[4^#(x₁)]	=	x₁ + 32279
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	32279
[5^#(x₁)]	=	x₁ + 8838
[2(x₁)]	=	23442

together with the usable rules

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

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

5^#(5(0(4(x1))))	→	5^#(5(1(0(4(x1)))))	(162)
3^#(0(5(0(x1))))	→	5^#(0(3(0(1(1(x1))))))	(96)
5^#(5(4(3(x1))))	→	3^#(4(5(5(1(x1)))))	(54)

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 2 components.

The 1^st component contains the pair

5^#(1(0(3(x1))))

→

5^#(3(0(1(1(1(x1))))))

(190)

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(x₁)]

1	0
0	0

· x₁ +

[4(x₁)]

[5(x₁)]

1	0
1	0

· x₁ +

[3(x₁)]

x₁ +

[2^#(x₁)]

[4^#(x₁)]

[0(x₁)]

0	1
0	1

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

1	0
0	0

· x₁ +

[2(x₁)]

together with the usable rules

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

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

5^#(1(0(3(x1))))

→

5^#(3(0(1(1(1(x1))))))

(190)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

3^#(2(1(4(x1))))	→	3^#(1(4(x1)))	(156)
3^#(0(4(0(x1))))	→	3^#(0(0(1(1(4(x1))))))	(113)
3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

[1(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

0	1
0	0

· x₁ +

[5(x₁)]

[3(x₁)]

0	1
0	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

[3^#(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

[2(x₁)]

0	0
0	1

· x₁ +

together with the usable rules

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

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

3^#(0(0(3(x1))))	→	3^#(0(3(0(1(1(x1))))))	(55)
3^#(1(0(4(x1))))	→	3^#(4(0(x1)))	(104)
3^#(2(1(4(x1))))	→	3^#(x1)	(71)
3^#(1(0(0(x1))))	→	3^#(0(0(x1)))	(140)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

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

(113)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 32278
[1(x₁)]	=	0
[4(x₁)]	=	1
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	0
[2^#(x₁)]	=	0
[4^#(x₁)]	=	x₁ + 32279
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	x₁ + 32279
[5^#(x₁)]	=	8838
[2(x₁)]	=	1

together with the usable rules

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

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

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

→

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

(113)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

4^#(4(5(0(0(x1)))))	→	4^#(4(0(1(5(0(x1))))))	(121)
4^#(0(1(0(3(x1)))))	→	4^#(0(3(0(1(1(x1))))))	(83)
4^#(2(5(0(4(x1)))))	→	4^#(4(1(0(5(2(x1))))))	(130)
4^#(2(3(5(4(x1)))))	→	4^#(4(5(3(x1))))	(175)

1.1.1.1.1.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 32278
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	1
[5(x₁)]	=	2
[3(x₁)]	=	2
[2^#(x₁)]	=	0
[4^#(x₁)]	=	x₁ + 32279
[0(x₁)]	=	x₁ + 0
[3^#(x₁)]	=	32279
[5^#(x₁)]	=	8838
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

4^#(2(5(0(4(x1)))))	→	4^#(4(1(0(5(2(x1))))))	(130)
4^#(2(3(5(4(x1)))))	→	4^#(4(5(3(x1))))	(175)

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

4^#(4(5(0(0(x1)))))	→	4^#(4(0(1(5(0(x1))))))	(121)
4^#(0(1(0(3(x1)))))	→	4^#(0(3(0(1(1(x1))))))	(83)

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)]

[1(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

1	0
1	0

· x₁ +

[3(x₁)]

1	0
1	1

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

0	1
0	0

· x₁ +

[0(x₁)]

1	1
0	0

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

[2(x₁)]

together with the usable rules

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

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

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

→

4^#(4(0(1(5(0(x1))))))

(121)

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

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

→

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

(83)

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(x₁)]

0	0
1	0

· x₁ +

[4(x₁)]

[5(x₁)]

[3(x₁)]

0	0
1	0

· x₁ +

[2^#(x₁)]

[4^#(x₁)]

0	1
1	0

· x₁ +

[0(x₁)]

1	1
0	1

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

[2(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

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

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

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

→

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

(83)

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 0 components.