Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/57355)

The rewrite relation of the following TRS is considered.

0(1(0(2(x1))))	→	2(0(3(1(0(x1)))))	(1)
0(1(0(2(x1))))	→	2(0(0(3(1(2(x1))))))	(2)
0(1(0(2(x1))))	→	2(0(3(1(0(4(x1))))))	(3)
0(1(0(2(x1))))	→	2(2(0(3(1(0(x1))))))	(4)
0(1(0(2(x1))))	→	2(3(1(0(0(2(x1))))))	(5)
0(1(0(2(x1))))	→	2(3(1(0(3(0(x1))))))	(6)
0(1(0(2(x1))))	→	4(1(0(3(0(2(x1))))))	(7)
0(1(0(2(x1))))	→	4(1(0(4(0(2(x1))))))	(8)
0(1(4(2(x1))))	→	2(3(1(0(4(x1)))))	(9)
0(1(4(2(x1))))	→	2(4(0(3(1(x1)))))	(10)
0(1(4(2(x1))))	→	3(2(1(0(4(x1)))))	(11)
0(1(4(2(x1))))	→	3(2(1(4(0(x1)))))	(12)
0(1(4(2(x1))))	→	4(0(3(1(2(x1)))))	(13)
0(1(4(2(x1))))	→	4(1(0(3(2(x1)))))	(14)
0(1(4(2(x1))))	→	4(1(0(4(2(x1)))))	(15)
0(1(4(2(x1))))	→	4(1(0(5(2(x1)))))	(16)
0(1(4(2(x1))))	→	2(0(3(1(0(4(x1))))))	(17)
0(1(4(2(x1))))	→	2(0(3(1(4(4(x1))))))	(18)
0(1(4(2(x1))))	→	2(3(1(4(0(4(x1))))))	(19)
0(1(4(2(x1))))	→	2(4(3(0(4(1(x1))))))	(20)
0(1(4(2(x1))))	→	2(4(3(1(0(3(x1))))))	(21)
0(1(4(2(x1))))	→	3(2(1(0(4(1(x1))))))	(22)
0(1(4(2(x1))))	→	3(2(2(1(4(0(x1))))))	(23)
0(1(4(2(x1))))	→	3(2(3(1(0(4(x1))))))	(24)
0(1(4(2(x1))))	→	3(2(3(1(4(0(x1))))))	(25)
0(1(4(2(x1))))	→	4(0(3(1(3(2(x1))))))	(26)
0(1(4(2(x1))))	→	4(0(3(1(4(2(x1))))))	(27)
0(1(4(2(x1))))	→	4(1(0(4(3(2(x1))))))	(28)
0(1(4(2(x1))))	→	4(1(0(4(5(2(x1))))))	(29)
0(1(4(2(x1))))	→	4(1(0(5(3(2(x1))))))	(30)
0(1(4(2(x1))))	→	4(1(1(0(5(2(x1))))))	(31)
0(1(4(2(x1))))	→	4(1(3(0(5(2(x1))))))	(32)
0(1(4(2(x1))))	→	4(3(0(3(1(2(x1))))))	(33)
0(1(4(2(x1))))	→	4(4(0(3(1(2(x1))))))	(34)
0(0(1(0(2(x1)))))	→	1(0(0(2(0(4(x1))))))	(35)
0(0(1(0(2(x1)))))	→	1(0(4(0(0(2(x1))))))	(36)
0(0(1(0(2(x1)))))	→	2(1(0(3(0(0(x1))))))	(37)
0(0(1(4(2(x1)))))	→	0(0(3(1(2(4(x1))))))	(38)
0(0(1(4(2(x1)))))	→	0(2(3(1(0(4(x1))))))	(39)
0(0(1(4(2(x1)))))	→	0(2(4(0(3(1(x1))))))	(40)
0(0(1(4(2(x1)))))	→	0(3(1(0(2(4(x1))))))	(41)
0(0(1(4(2(x1)))))	→	1(0(3(4(0(2(x1))))))	(42)
0(0(1(4(2(x1)))))	→	2(0(0(3(1(4(x1))))))	(43)
0(0(1(4(2(x1)))))	→	2(1(0(4(0(0(x1))))))	(44)
0(1(2(0(2(x1)))))	→	2(0(1(0(4(2(x1))))))	(45)
0(1(2(4(2(x1)))))	→	2(3(1(0(2(4(x1))))))	(46)
0(1(2(4(2(x1)))))	→	4(1(0(2(2(4(x1))))))	(47)
0(1(3(4(2(x1)))))	→	2(3(1(4(4(0(x1))))))	(48)
0(1(3(4(2(x1)))))	→	2(4(3(0(4(1(x1))))))	(49)
0(1(3(4(2(x1)))))	→	3(2(1(0(4(0(x1))))))	(50)
0(1(3(4(2(x1)))))	→	4(0(3(3(1(2(x1))))))	(51)
0(1(3(4(2(x1)))))	→	4(1(4(0(3(2(x1))))))	(52)
0(1(4(0(2(x1)))))	→	2(0(3(1(0(4(x1))))))	(53)
0(1(5(0(2(x1)))))	→	0(2(3(1(0(5(x1))))))	(54)
0(1(5(0(2(x1)))))	→	3(0(5(1(0(2(x1))))))	(55)
0(1(5(0(2(x1)))))	→	5(1(3(0(0(2(x1))))))	(56)
0(1(5(4(2(x1)))))	→	0(4(4(1(2(5(x1))))))	(57)
0(1(5(4(2(x1)))))	→	1(0(4(5(1(2(x1))))))	(58)
0(1(5(4(2(x1)))))	→	2(0(4(4(5(1(x1))))))	(59)
0(1(5(4(2(x1)))))	→	4(0(2(3(1(5(x1))))))	(60)
0(1(5(4(2(x1)))))	→	4(1(0(2(5(2(x1))))))	(61)
0(1(5(4(2(x1)))))	→	4(1(0(5(2(5(x1))))))	(62)
0(1(5(4(2(x1)))))	→	4(2(1(3(0(5(x1))))))	(63)
0(1(5(4(2(x1)))))	→	4(3(1(0(2(5(x1))))))	(64)
0(1(5(4(2(x1)))))	→	4(4(0(5(1(2(x1))))))	(65)
0(1(5(4(2(x1)))))	→	4(4(2(1(0(5(x1))))))	(66)
0(1(5(4(2(x1)))))	→	5(0(4(5(2(1(x1))))))	(67)
0(1(5(4(2(x1)))))	→	5(1(2(0(4(3(x1))))))	(68)
0(1(5(4(2(x1)))))	→	5(3(1(0(4(2(x1))))))	(69)
0(2(1(4(2(x1)))))	→	0(4(4(1(2(2(x1))))))	(70)
0(2(1(4(2(x1)))))	→	3(2(2(1(4(0(x1))))))	(71)
0(2(1(4(2(x1)))))	→	4(1(0(3(2(2(x1))))))	(72)
5(0(1(4(2(x1)))))	→	2(0(4(3(5(1(x1))))))	(73)
5(0(1(4(2(x1)))))	→	2(4(0(3(1(5(x1))))))	(74)
5(0(1(4(2(x1)))))	→	4(1(0(5(3(2(x1))))))	(75)
5(0(1(4(2(x1)))))	→	5(2(1(1(0(4(x1))))))	(76)
5(0(2(0(2(x1)))))	→	5(0(3(0(2(2(x1))))))	(77)
5(0(2(0(2(x1)))))	→	5(0(4(0(2(2(x1))))))	(78)
5(0(2(4(2(x1)))))	→	5(4(0(3(2(2(x1))))))	(79)
5(1(5(0(2(x1)))))	→	5(2(1(4(5(0(x1))))))	(80)
5(1(5(4(2(x1)))))	→	5(2(1(0(4(5(x1))))))	(81)
5(4(1(4(2(x1)))))	→	3(2(1(4(4(5(x1))))))	(82)
5(4(1(4(2(x1)))))	→	4(4(3(5(1(2(x1))))))	(83)
5(4(2(0(2(x1)))))	→	3(0(5(2(2(4(x1))))))	(84)
5(4(2(0(2(x1)))))	→	4(0(5(3(2(2(x1))))))	(85)
5(4(2(0(2(x1)))))	→	5(2(2(2(4(0(x1))))))	(86)
5(4(2(0(2(x1)))))	→	5(3(2(2(4(0(x1))))))	(87)
5(4(2(0(2(x1)))))	→	5(4(2(2(4(0(x1))))))	(88)
5(4(2(4(2(x1)))))	→	0(4(4(5(2(2(x1))))))	(89)
5(4(2(4(2(x1)))))	→	5(4(4(3(2(2(x1))))))	(90)
5(4(5(4(2(x1)))))	→	4(5(0(4(5(2(x1))))))	(91)

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^#(0(1(4(2(x1)))))	→	0^#(0(x1))	(169)
5^#(4(2(0(2(x1)))))	→	0^#(x1)	(108)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(1(0(2(x1))))	→	0^#(0(2(x1)))	(162)
0^#(1(5(4(2(x1)))))	→	0^#(2(5(x1)))	(218)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
0^#(1(5(0(2(x1)))))	→	0^#(0(2(x1)))	(161)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(2(1(4(2(x1)))))	→	0^#(x1)	(158)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
5^#(4(2(0(2(x1)))))	→	0^#(x1)	(108)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(0(1(4(2(x1)))))	→	0^#(x1)	(146)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(0(1(0(2(x1)))))	→	0^#(0(x1))	(210)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
5^#(0(1(4(2(x1)))))	→	5^#(x1)	(209)
5^#(1(5(4(2(x1)))))	→	5^#(x1)	(207)
5^#(4(1(4(2(x1)))))	→	5^#(x1)	(205)
0^#(0(1(0(2(x1)))))	→	0^#(0(2(x1)))	(201)
0^#(1(5(4(2(x1)))))	→	0^#(5(x1))	(140)
0^#(1(3(4(2(x1)))))	→	0^#(x1)	(193)
0^#(0(1(4(2(x1)))))	→	0^#(2(x1))	(199)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
5^#(1(5(0(2(x1)))))	→	0^#(x1)	(128)
5^#(1(5(0(2(x1)))))	→	5^#(0(x1))	(126)
5^#(0(1(4(2(x1)))))	→	5^#(1(x1))	(194)
0^#(1(3(4(2(x1)))))	→	0^#(x1)	(193)
0^#(1(5(0(2(x1)))))	→	5^#(1(0(2(x1))))	(116)
0^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(114)
0^#(1(5(4(2(x1)))))	→	0^#(5(x1))	(140)
0^#(0(1(0(2(x1)))))	→	0^#(x1)	(189)
0^#(1(5(0(2(x1)))))	→	5^#(x1)	(187)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(1(5(0(2(x1)))))	→	0^#(5(1(0(2(x1)))))	(111)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
5^#(4(2(0(2(x1)))))	→	0^#(x1)	(108)
0^#(1(5(4(2(x1)))))	→	5^#(1(x1))	(99)

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	1
0	0

· x₁ +

[1(x₁)]

0	0
1	1

· x₁ +

[4(x₁)]

0	1
0	0

· x₁ +

[5(x₁)]

1	1
0	0

· x₁ +

25496

[3(x₁)]

1	0
0	0

· x₁ +

[0(x₁)]

0	1
0	0

· x₁ +

[5^#(x₁)]

1	1
0	0

· x₁ +

25495

[2(x₁)]

0	0
1	1

· x₁ +

together with the usable rules

0(1(4(2(x1))))	→	2(0(3(1(4(4(x1))))))	(18)
0(1(3(4(2(x1)))))	→	3(2(1(0(4(0(x1))))))	(50)
5(1(5(0(2(x1)))))	→	5(2(1(4(5(0(x1))))))	(80)
0(1(0(2(x1))))	→	2(2(0(3(1(0(x1))))))	(4)
0(1(4(2(x1))))	→	4(1(0(4(2(x1)))))	(15)
0(1(0(2(x1))))	→	4(1(0(4(0(2(x1))))))	(8)
0(1(5(0(2(x1)))))	→	0(2(3(1(0(5(x1))))))	(54)
0(1(0(2(x1))))	→	2(0(3(1(0(x1)))))	(1)
5(0(2(0(2(x1)))))	→	5(0(3(0(2(2(x1))))))	(77)
0(1(0(2(x1))))	→	2(0(3(1(0(4(x1))))))	(3)
0(1(4(2(x1))))	→	4(1(0(5(2(x1)))))	(16)
0(1(4(2(x1))))	→	2(4(3(1(0(3(x1))))))	(21)
0(0(1(0(2(x1)))))	→	1(0(4(0(0(2(x1))))))	(36)
0(1(5(4(2(x1)))))	→	5(1(2(0(4(3(x1))))))	(68)
5(4(2(0(2(x1)))))	→	4(0(5(3(2(2(x1))))))	(85)
0(1(4(2(x1))))	→	4(0(3(1(3(2(x1))))))	(26)
0(1(5(4(2(x1)))))	→	4(2(1(3(0(5(x1))))))	(63)
0(1(4(2(x1))))	→	2(3(1(4(0(4(x1))))))	(19)
0(1(4(2(x1))))	→	4(1(3(0(5(2(x1))))))	(32)
0(1(4(2(x1))))	→	2(0(3(1(0(4(x1))))))	(17)
0(1(5(4(2(x1)))))	→	4(0(2(3(1(5(x1))))))	(60)
0(1(4(2(x1))))	→	4(0(3(1(4(2(x1))))))	(27)
5(4(2(0(2(x1)))))	→	5(3(2(2(4(0(x1))))))	(87)
5(4(2(0(2(x1)))))	→	3(0(5(2(2(4(x1))))))	(84)
0(1(4(2(x1))))	→	4(4(0(3(1(2(x1))))))	(34)
0(1(4(2(x1))))	→	3(2(1(0(4(1(x1))))))	(22)
0(1(4(2(x1))))	→	4(1(0(4(3(2(x1))))))	(28)
0(1(5(4(2(x1)))))	→	4(4(0(5(1(2(x1))))))	(65)
0(0(1(4(2(x1)))))	→	2(1(0(4(0(0(x1))))))	(44)
0(1(0(2(x1))))	→	2(3(1(0(0(2(x1))))))	(5)
0(2(1(4(2(x1)))))	→	4(1(0(3(2(2(x1))))))	(72)
0(1(4(2(x1))))	→	4(3(0(3(1(2(x1))))))	(33)
0(1(5(4(2(x1)))))	→	4(3(1(0(2(5(x1))))))	(64)
0(1(4(2(x1))))	→	2(4(0(3(1(x1)))))	(10)
0(0(1(4(2(x1)))))	→	0(2(3(1(0(4(x1))))))	(39)
0(1(0(2(x1))))	→	4(1(0(3(0(2(x1))))))	(7)
5(4(2(0(2(x1)))))	→	5(4(2(2(4(0(x1))))))	(88)
0(1(4(2(x1))))	→	2(4(3(0(4(1(x1))))))	(20)
0(1(4(2(x1))))	→	3(2(3(1(4(0(x1))))))	(25)
0(1(3(4(2(x1)))))	→	2(4(3(0(4(1(x1))))))	(49)
0(1(3(4(2(x1)))))	→	4(1(4(0(3(2(x1))))))	(52)
0(1(4(2(x1))))	→	4(1(0(5(3(2(x1))))))	(30)
0(1(5(4(2(x1)))))	→	4(1(0(5(2(5(x1))))))	(62)
0(1(4(2(x1))))	→	4(1(0(3(2(x1)))))	(14)
5(4(1(4(2(x1)))))	→	3(2(1(4(4(5(x1))))))	(82)
5(4(2(4(2(x1)))))	→	0(4(4(5(2(2(x1))))))	(89)
0(1(5(0(2(x1)))))	→	5(1(3(0(0(2(x1))))))	(56)
5(0(2(4(2(x1)))))	→	5(4(0(3(2(2(x1))))))	(79)
0(1(4(2(x1))))	→	4(1(1(0(5(2(x1))))))	(31)
0(1(4(2(x1))))	→	3(2(1(4(0(x1)))))	(12)
0(1(5(4(2(x1)))))	→	5(3(1(0(4(2(x1))))))	(69)
0(1(2(0(2(x1)))))	→	2(0(1(0(4(2(x1))))))	(45)
5(0(2(0(2(x1)))))	→	5(0(4(0(2(2(x1))))))	(78)
5(1(5(4(2(x1)))))	→	5(2(1(0(4(5(x1))))))	(81)
0(1(4(2(x1))))	→	3(2(2(1(4(0(x1))))))	(23)
0(2(1(4(2(x1)))))	→	0(4(4(1(2(2(x1))))))	(70)
0(1(4(2(x1))))	→	3(2(3(1(0(4(x1))))))	(24)
5(0(1(4(2(x1)))))	→	5(2(1(1(0(4(x1))))))	(76)
0(1(5(4(2(x1)))))	→	0(4(4(1(2(5(x1))))))	(57)
0(1(4(2(x1))))	→	3(2(1(0(4(x1)))))	(11)
0(1(4(2(x1))))	→	2(3(1(0(4(x1)))))	(9)
0(1(4(2(x1))))	→	4(0(3(1(2(x1)))))	(13)
0(1(3(4(2(x1)))))	→	4(0(3(3(1(2(x1))))))	(51)
5(4(2(4(2(x1)))))	→	5(4(4(3(2(2(x1))))))	(90)
0(0(1(4(2(x1)))))	→	0(2(4(0(3(1(x1))))))	(40)
0(1(5(4(2(x1)))))	→	5(0(4(5(2(1(x1))))))	(67)
0(1(5(0(2(x1)))))	→	3(0(5(1(0(2(x1))))))	(55)
0(1(5(4(2(x1)))))	→	2(0(4(4(5(1(x1))))))	(59)
0(1(0(2(x1))))	→	2(3(1(0(3(0(x1))))))	(6)
0(0(1(4(2(x1)))))	→	0(0(3(1(2(4(x1))))))	(38)
0(1(5(4(2(x1)))))	→	4(1(0(2(5(2(x1))))))	(61)
0(1(5(4(2(x1)))))	→	1(0(4(5(1(2(x1))))))	(58)
5(0(1(4(2(x1)))))	→	2(4(0(3(1(5(x1))))))	(74)
5(0(1(4(2(x1)))))	→	4(1(0(5(3(2(x1))))))	(75)
0(1(3(4(2(x1)))))	→	2(3(1(4(4(0(x1))))))	(48)
0(2(1(4(2(x1)))))	→	3(2(2(1(4(0(x1))))))	(71)
0(1(4(0(2(x1)))))	→	2(0(3(1(0(4(x1))))))	(53)
0(1(2(4(2(x1)))))	→	4(1(0(2(2(4(x1))))))	(47)
5(0(1(4(2(x1)))))	→	2(0(4(3(5(1(x1))))))	(73)
5(4(5(4(2(x1)))))	→	4(5(0(4(5(2(x1))))))	(91)
0(0(1(0(2(x1)))))	→	2(1(0(3(0(0(x1))))))	(37)
0(0(1(4(2(x1)))))	→	0(3(1(0(2(4(x1))))))	(41)
0(0(1(4(2(x1)))))	→	1(0(3(4(0(2(x1))))))	(42)
0(1(2(4(2(x1)))))	→	2(3(1(0(2(4(x1))))))	(46)
0(1(5(4(2(x1)))))	→	4(4(2(1(0(5(x1))))))	(66)
5(4(1(4(2(x1)))))	→	4(4(3(5(1(2(x1))))))	(83)
0(0(1(0(2(x1)))))	→	1(0(0(2(0(4(x1))))))	(35)
0(1(4(2(x1))))	→	4(1(0(4(5(2(x1))))))	(29)
0(0(1(4(2(x1)))))	→	2(0(0(3(1(4(x1))))))	(43)
5(4(2(0(2(x1)))))	→	5(2(2(2(4(0(x1))))))	(86)
0(1(0(2(x1))))	→	2(0(0(3(1(2(x1))))))	(2)

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

5^#(4(2(0(2(x1)))))	→	0^#(x1)	(108)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(1(5(0(2(x1)))))	→	0^#(0(2(x1)))	(161)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
5^#(4(2(0(2(x1)))))	→	0^#(x1)	(108)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
5^#(1(5(4(2(x1)))))	→	5^#(x1)	(207)
0^#(1(5(4(2(x1)))))	→	5^#(x1)	(151)
5^#(1(5(0(2(x1)))))	→	0^#(x1)	(128)
5^#(1(5(0(2(x1)))))	→	5^#(0(x1))	(126)
0^#(1(5(0(2(x1)))))	→	5^#(1(0(2(x1))))	(116)
0^#(1(5(0(2(x1)))))	→	5^#(x1)	(187)
5^#(4(2(0(2(x1)))))	→	0^#(x1)	(108)
0^#(1(5(4(2(x1)))))	→	5^#(1(x1))	(99)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

5^#(4(1(4(2(x1)))))	→	5^#(x1)	(205)
5^#(0(1(4(2(x1)))))	→	5^#(x1)	(209)

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
1	0

· x₁ +

[4(x₁)]

1	0
0	0

· x₁ +

[5(x₁)]

25302

[3(x₁)]

[0(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

1	0
1	0

· x₁ +

25495

[2(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

0(1(4(2(x1))))	→	2(0(3(1(4(4(x1))))))	(18)
0(1(3(4(2(x1)))))	→	3(2(1(0(4(0(x1))))))	(50)
5(1(5(0(2(x1)))))	→	5(2(1(4(5(0(x1))))))	(80)
0(1(0(2(x1))))	→	2(2(0(3(1(0(x1))))))	(4)
0(1(4(2(x1))))	→	4(1(0(4(2(x1)))))	(15)
0(1(0(2(x1))))	→	4(1(0(4(0(2(x1))))))	(8)
0(1(5(0(2(x1)))))	→	0(2(3(1(0(5(x1))))))	(54)
0(1(0(2(x1))))	→	2(0(3(1(0(x1)))))	(1)
5(0(2(0(2(x1)))))	→	5(0(3(0(2(2(x1))))))	(77)
0(1(0(2(x1))))	→	2(0(3(1(0(4(x1))))))	(3)
0(1(4(2(x1))))	→	4(1(0(5(2(x1)))))	(16)
0(1(4(2(x1))))	→	2(4(3(1(0(3(x1))))))	(21)
0(0(1(0(2(x1)))))	→	1(0(4(0(0(2(x1))))))	(36)
0(1(5(4(2(x1)))))	→	5(1(2(0(4(3(x1))))))	(68)
5(4(2(0(2(x1)))))	→	4(0(5(3(2(2(x1))))))	(85)
0(1(4(2(x1))))	→	4(0(3(1(3(2(x1))))))	(26)
0(1(5(4(2(x1)))))	→	4(2(1(3(0(5(x1))))))	(63)
0(1(4(2(x1))))	→	2(3(1(4(0(4(x1))))))	(19)
0(1(4(2(x1))))	→	4(1(3(0(5(2(x1))))))	(32)
0(1(4(2(x1))))	→	2(0(3(1(0(4(x1))))))	(17)
0(1(5(4(2(x1)))))	→	4(0(2(3(1(5(x1))))))	(60)
0(1(4(2(x1))))	→	4(0(3(1(4(2(x1))))))	(27)
5(4(2(0(2(x1)))))	→	5(3(2(2(4(0(x1))))))	(87)
5(4(2(0(2(x1)))))	→	3(0(5(2(2(4(x1))))))	(84)
0(1(4(2(x1))))	→	4(4(0(3(1(2(x1))))))	(34)
0(1(4(2(x1))))	→	3(2(1(0(4(1(x1))))))	(22)
0(1(4(2(x1))))	→	4(1(0(4(3(2(x1))))))	(28)
0(1(5(4(2(x1)))))	→	4(4(0(5(1(2(x1))))))	(65)
0(0(1(4(2(x1)))))	→	2(1(0(4(0(0(x1))))))	(44)
0(1(0(2(x1))))	→	2(3(1(0(0(2(x1))))))	(5)
0(2(1(4(2(x1)))))	→	4(1(0(3(2(2(x1))))))	(72)
0(1(4(2(x1))))	→	4(3(0(3(1(2(x1))))))	(33)
0(1(5(4(2(x1)))))	→	4(3(1(0(2(5(x1))))))	(64)
0(1(4(2(x1))))	→	2(4(0(3(1(x1)))))	(10)
0(0(1(4(2(x1)))))	→	0(2(3(1(0(4(x1))))))	(39)
0(1(0(2(x1))))	→	4(1(0(3(0(2(x1))))))	(7)
5(4(2(0(2(x1)))))	→	5(4(2(2(4(0(x1))))))	(88)
0(1(4(2(x1))))	→	2(4(3(0(4(1(x1))))))	(20)
0(1(4(2(x1))))	→	3(2(3(1(4(0(x1))))))	(25)
0(1(3(4(2(x1)))))	→	2(4(3(0(4(1(x1))))))	(49)
0(1(3(4(2(x1)))))	→	4(1(4(0(3(2(x1))))))	(52)
0(1(4(2(x1))))	→	4(1(0(5(3(2(x1))))))	(30)
0(1(5(4(2(x1)))))	→	4(1(0(5(2(5(x1))))))	(62)
0(1(4(2(x1))))	→	4(1(0(3(2(x1)))))	(14)
5(4(1(4(2(x1)))))	→	3(2(1(4(4(5(x1))))))	(82)
5(4(2(4(2(x1)))))	→	0(4(4(5(2(2(x1))))))	(89)
0(1(5(0(2(x1)))))	→	5(1(3(0(0(2(x1))))))	(56)
5(0(2(4(2(x1)))))	→	5(4(0(3(2(2(x1))))))	(79)
0(1(4(2(x1))))	→	4(1(1(0(5(2(x1))))))	(31)
0(1(4(2(x1))))	→	3(2(1(4(0(x1)))))	(12)
0(1(5(4(2(x1)))))	→	5(3(1(0(4(2(x1))))))	(69)
0(1(2(0(2(x1)))))	→	2(0(1(0(4(2(x1))))))	(45)
5(0(2(0(2(x1)))))	→	5(0(4(0(2(2(x1))))))	(78)
5(1(5(4(2(x1)))))	→	5(2(1(0(4(5(x1))))))	(81)
0(1(4(2(x1))))	→	3(2(2(1(4(0(x1))))))	(23)
0(2(1(4(2(x1)))))	→	0(4(4(1(2(2(x1))))))	(70)
0(1(4(2(x1))))	→	3(2(3(1(0(4(x1))))))	(24)
5(0(1(4(2(x1)))))	→	5(2(1(1(0(4(x1))))))	(76)
0(1(5(4(2(x1)))))	→	0(4(4(1(2(5(x1))))))	(57)
0(1(4(2(x1))))	→	3(2(1(0(4(x1)))))	(11)
0(1(4(2(x1))))	→	2(3(1(0(4(x1)))))	(9)
0(1(4(2(x1))))	→	4(0(3(1(2(x1)))))	(13)
0(1(3(4(2(x1)))))	→	4(0(3(3(1(2(x1))))))	(51)
5(4(2(4(2(x1)))))	→	5(4(4(3(2(2(x1))))))	(90)
0(0(1(4(2(x1)))))	→	0(2(4(0(3(1(x1))))))	(40)
0(1(5(4(2(x1)))))	→	5(0(4(5(2(1(x1))))))	(67)
0(1(5(0(2(x1)))))	→	3(0(5(1(0(2(x1))))))	(55)
0(1(5(4(2(x1)))))	→	2(0(4(4(5(1(x1))))))	(59)
0(1(0(2(x1))))	→	2(3(1(0(3(0(x1))))))	(6)
0(0(1(4(2(x1)))))	→	0(0(3(1(2(4(x1))))))	(38)
0(1(5(4(2(x1)))))	→	4(1(0(2(5(2(x1))))))	(61)
0(1(5(4(2(x1)))))	→	1(0(4(5(1(2(x1))))))	(58)
5(0(1(4(2(x1)))))	→	2(4(0(3(1(5(x1))))))	(74)
5(0(1(4(2(x1)))))	→	4(1(0(5(3(2(x1))))))	(75)
0(1(3(4(2(x1)))))	→	2(3(1(4(4(0(x1))))))	(48)
0(2(1(4(2(x1)))))	→	3(2(2(1(4(0(x1))))))	(71)
0(1(4(0(2(x1)))))	→	2(0(3(1(0(4(x1))))))	(53)
0(1(2(4(2(x1)))))	→	4(1(0(2(2(4(x1))))))	(47)
5(0(1(4(2(x1)))))	→	2(0(4(3(5(1(x1))))))	(73)
5(4(5(4(2(x1)))))	→	4(5(0(4(5(2(x1))))))	(91)
0(0(1(0(2(x1)))))	→	2(1(0(3(0(0(x1))))))	(37)
0(0(1(4(2(x1)))))	→	0(3(1(0(2(4(x1))))))	(41)
0(0(1(4(2(x1)))))	→	1(0(3(4(0(2(x1))))))	(42)
0(1(2(4(2(x1)))))	→	2(3(1(0(2(4(x1))))))	(46)
0(1(5(4(2(x1)))))	→	4(4(2(1(0(5(x1))))))	(66)
5(4(1(4(2(x1)))))	→	4(4(3(5(1(2(x1))))))	(83)
0(0(1(0(2(x1)))))	→	1(0(0(2(0(4(x1))))))	(35)
0(1(4(2(x1))))	→	4(1(0(4(5(2(x1))))))	(29)
0(0(1(4(2(x1)))))	→	2(0(0(3(1(4(x1))))))	(43)
5(4(2(0(2(x1)))))	→	5(2(2(2(4(0(x1))))))	(86)
0(1(0(2(x1))))	→	2(0(0(3(1(2(x1))))))	(2)

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

5^#(4(1(4(2(x1)))))	→	5^#(x1)	(205)
5^#(0(1(4(2(x1)))))	→	5^#(x1)	(209)

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

0^#(1(3(4(2(x1)))))	→	0^#(x1)	(193)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
0^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(114)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
0^#(0(1(0(2(x1)))))	→	0^#(0(2(x1)))	(201)
0^#(1(5(4(2(x1)))))	→	0^#(5(x1))	(140)
0^#(0(1(4(2(x1)))))	→	0^#(x1)	(146)
0^#(0(1(4(2(x1)))))	→	0^#(0(x1))	(169)
0^#(1(0(2(x1))))	→	0^#(0(2(x1)))	(162)
0^#(1(5(4(2(x1)))))	→	0^#(2(5(x1)))	(218)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(1(5(0(2(x1)))))	→	0^#(5(1(0(2(x1)))))	(111)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
0^#(1(3(4(2(x1)))))	→	0^#(x1)	(193)
0^#(2(1(4(2(x1)))))	→	0^#(x1)	(158)
0^#(0(1(0(2(x1)))))	→	0^#(x1)	(189)
0^#(0(1(0(2(x1)))))	→	0^#(0(x1))	(210)
0^#(0(1(4(2(x1)))))	→	0^#(2(x1))	(199)
0^#(1(5(4(2(x1)))))	→	0^#(5(x1))	(140)

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
1	0

· x₁ +

113

56471

[4(x₁)]

0	1
0	0

· x₁ +

[5(x₁)]

69447

12974

[3(x₁)]

1	0
0	0

· x₁ +

[0(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

25495

[2(x₁)]

0	0
0	1

· x₁ +

25949

12974

together with the usable rules

0(1(4(2(x1))))	→	2(0(3(1(4(4(x1))))))	(18)
0(1(3(4(2(x1)))))	→	3(2(1(0(4(0(x1))))))	(50)
5(1(5(0(2(x1)))))	→	5(2(1(4(5(0(x1))))))	(80)
0(1(0(2(x1))))	→	2(2(0(3(1(0(x1))))))	(4)
0(1(4(2(x1))))	→	4(1(0(4(2(x1)))))	(15)
0(1(0(2(x1))))	→	4(1(0(4(0(2(x1))))))	(8)
0(1(5(0(2(x1)))))	→	0(2(3(1(0(5(x1))))))	(54)
0(1(0(2(x1))))	→	2(0(3(1(0(x1)))))	(1)
5(0(2(0(2(x1)))))	→	5(0(3(0(2(2(x1))))))	(77)
0(1(0(2(x1))))	→	2(0(3(1(0(4(x1))))))	(3)
0(1(4(2(x1))))	→	4(1(0(5(2(x1)))))	(16)
0(1(4(2(x1))))	→	2(4(3(1(0(3(x1))))))	(21)
0(0(1(0(2(x1)))))	→	1(0(4(0(0(2(x1))))))	(36)
0(1(5(4(2(x1)))))	→	5(1(2(0(4(3(x1))))))	(68)
5(4(2(0(2(x1)))))	→	4(0(5(3(2(2(x1))))))	(85)
0(1(4(2(x1))))	→	4(0(3(1(3(2(x1))))))	(26)
0(1(5(4(2(x1)))))	→	4(2(1(3(0(5(x1))))))	(63)
0(1(4(2(x1))))	→	2(3(1(4(0(4(x1))))))	(19)
0(1(4(2(x1))))	→	4(1(3(0(5(2(x1))))))	(32)
0(1(4(2(x1))))	→	2(0(3(1(0(4(x1))))))	(17)
0(1(5(4(2(x1)))))	→	4(0(2(3(1(5(x1))))))	(60)
0(1(4(2(x1))))	→	4(0(3(1(4(2(x1))))))	(27)
5(4(2(0(2(x1)))))	→	5(3(2(2(4(0(x1))))))	(87)
5(4(2(0(2(x1)))))	→	3(0(5(2(2(4(x1))))))	(84)
0(1(4(2(x1))))	→	4(4(0(3(1(2(x1))))))	(34)
0(1(4(2(x1))))	→	3(2(1(0(4(1(x1))))))	(22)
0(1(4(2(x1))))	→	4(1(0(4(3(2(x1))))))	(28)
0(1(5(4(2(x1)))))	→	4(4(0(5(1(2(x1))))))	(65)
0(0(1(4(2(x1)))))	→	2(1(0(4(0(0(x1))))))	(44)
0(1(0(2(x1))))	→	2(3(1(0(0(2(x1))))))	(5)
0(2(1(4(2(x1)))))	→	4(1(0(3(2(2(x1))))))	(72)
0(1(4(2(x1))))	→	4(3(0(3(1(2(x1))))))	(33)
0(1(5(4(2(x1)))))	→	4(3(1(0(2(5(x1))))))	(64)
0(1(4(2(x1))))	→	2(4(0(3(1(x1)))))	(10)
0(0(1(4(2(x1)))))	→	0(2(3(1(0(4(x1))))))	(39)
0(1(0(2(x1))))	→	4(1(0(3(0(2(x1))))))	(7)
5(4(2(0(2(x1)))))	→	5(4(2(2(4(0(x1))))))	(88)
0(1(4(2(x1))))	→	2(4(3(0(4(1(x1))))))	(20)
0(1(4(2(x1))))	→	3(2(3(1(4(0(x1))))))	(25)
0(1(3(4(2(x1)))))	→	2(4(3(0(4(1(x1))))))	(49)
0(1(3(4(2(x1)))))	→	4(1(4(0(3(2(x1))))))	(52)
0(1(4(2(x1))))	→	4(1(0(5(3(2(x1))))))	(30)
0(1(5(4(2(x1)))))	→	4(1(0(5(2(5(x1))))))	(62)
0(1(4(2(x1))))	→	4(1(0(3(2(x1)))))	(14)
5(4(1(4(2(x1)))))	→	3(2(1(4(4(5(x1))))))	(82)
5(4(2(4(2(x1)))))	→	0(4(4(5(2(2(x1))))))	(89)
0(1(5(0(2(x1)))))	→	5(1(3(0(0(2(x1))))))	(56)
5(0(2(4(2(x1)))))	→	5(4(0(3(2(2(x1))))))	(79)
0(1(4(2(x1))))	→	4(1(1(0(5(2(x1))))))	(31)
0(1(4(2(x1))))	→	3(2(1(4(0(x1)))))	(12)
0(1(5(4(2(x1)))))	→	5(3(1(0(4(2(x1))))))	(69)
0(1(2(0(2(x1)))))	→	2(0(1(0(4(2(x1))))))	(45)
5(0(2(0(2(x1)))))	→	5(0(4(0(2(2(x1))))))	(78)
5(1(5(4(2(x1)))))	→	5(2(1(0(4(5(x1))))))	(81)
0(1(4(2(x1))))	→	3(2(2(1(4(0(x1))))))	(23)
0(2(1(4(2(x1)))))	→	0(4(4(1(2(2(x1))))))	(70)
0(1(4(2(x1))))	→	3(2(3(1(0(4(x1))))))	(24)
5(0(1(4(2(x1)))))	→	5(2(1(1(0(4(x1))))))	(76)
0(1(5(4(2(x1)))))	→	0(4(4(1(2(5(x1))))))	(57)
0(1(4(2(x1))))	→	3(2(1(0(4(x1)))))	(11)
0(1(4(2(x1))))	→	2(3(1(0(4(x1)))))	(9)
0(1(4(2(x1))))	→	4(0(3(1(2(x1)))))	(13)
0(1(3(4(2(x1)))))	→	4(0(3(3(1(2(x1))))))	(51)
5(4(2(4(2(x1)))))	→	5(4(4(3(2(2(x1))))))	(90)
0(0(1(4(2(x1)))))	→	0(2(4(0(3(1(x1))))))	(40)
0(1(5(4(2(x1)))))	→	5(0(4(5(2(1(x1))))))	(67)
0(1(5(0(2(x1)))))	→	3(0(5(1(0(2(x1))))))	(55)
0(1(5(4(2(x1)))))	→	2(0(4(4(5(1(x1))))))	(59)
0(1(0(2(x1))))	→	2(3(1(0(3(0(x1))))))	(6)
0(0(1(4(2(x1)))))	→	0(0(3(1(2(4(x1))))))	(38)
0(1(5(4(2(x1)))))	→	4(1(0(2(5(2(x1))))))	(61)
0(1(5(4(2(x1)))))	→	1(0(4(5(1(2(x1))))))	(58)
5(0(1(4(2(x1)))))	→	2(4(0(3(1(5(x1))))))	(74)
5(0(1(4(2(x1)))))	→	4(1(0(5(3(2(x1))))))	(75)
0(1(3(4(2(x1)))))	→	2(3(1(4(4(0(x1))))))	(48)
0(2(1(4(2(x1)))))	→	3(2(2(1(4(0(x1))))))	(71)
0(1(4(0(2(x1)))))	→	2(0(3(1(0(4(x1))))))	(53)
0(1(2(4(2(x1)))))	→	4(1(0(2(2(4(x1))))))	(47)
5(0(1(4(2(x1)))))	→	2(0(4(3(5(1(x1))))))	(73)
5(4(5(4(2(x1)))))	→	4(5(0(4(5(2(x1))))))	(91)
0(0(1(0(2(x1)))))	→	2(1(0(3(0(0(x1))))))	(37)
0(0(1(4(2(x1)))))	→	0(3(1(0(2(4(x1))))))	(41)
0(0(1(4(2(x1)))))	→	1(0(3(4(0(2(x1))))))	(42)
0(1(2(4(2(x1)))))	→	2(3(1(0(2(4(x1))))))	(46)
0(1(5(4(2(x1)))))	→	4(4(2(1(0(5(x1))))))	(66)
5(4(1(4(2(x1)))))	→	4(4(3(5(1(2(x1))))))	(83)
0(0(1(0(2(x1)))))	→	1(0(0(2(0(4(x1))))))	(35)
0(1(4(2(x1))))	→	4(1(0(4(5(2(x1))))))	(29)
0(0(1(4(2(x1)))))	→	2(0(0(3(1(4(x1))))))	(43)
5(4(2(0(2(x1)))))	→	5(2(2(2(4(0(x1))))))	(86)
0(1(0(2(x1))))	→	2(0(0(3(1(2(x1))))))	(2)

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

0^#(1(3(4(2(x1)))))	→	0^#(x1)	(193)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
0^#(1(5(0(2(x1)))))	→	0^#(5(x1))	(114)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
0^#(0(1(0(2(x1)))))	→	0^#(0(2(x1)))	(201)
0^#(1(5(4(2(x1)))))	→	0^#(5(x1))	(140)
0^#(0(1(4(2(x1)))))	→	0^#(x1)	(146)
0^#(0(1(4(2(x1)))))	→	0^#(0(x1))	(169)
0^#(1(0(2(x1))))	→	0^#(0(2(x1)))	(162)
0^#(1(5(4(2(x1)))))	→	0^#(2(5(x1)))	(218)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(1(4(2(x1))))	→	0^#(x1)	(145)
0^#(1(5(0(2(x1)))))	→	0^#(5(1(0(2(x1)))))	(111)
0^#(1(0(2(x1))))	→	0^#(x1)	(186)
0^#(1(3(4(2(x1)))))	→	0^#(x1)	(193)
0^#(2(1(4(2(x1)))))	→	0^#(x1)	(158)
0^#(0(1(0(2(x1)))))	→	0^#(x1)	(189)
0^#(0(1(0(2(x1)))))	→	0^#(0(x1))	(210)
0^#(0(1(4(2(x1)))))	→	0^#(2(x1))	(199)
0^#(1(5(4(2(x1)))))	→	0^#(5(x1))	(140)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.