Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212094)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

5^#(1(2(4(0(x1)))))	→	5^#(0(2(1(3(4(x1))))))	(51)
0^#(1(2(0(4(x1)))))	→	0^#(x1)	(52)
0^#(1(2(4(x1))))	→	5^#(0(2(3(1(x1)))))	(53)
5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
5^#(4(2(0(x1))))	→	0^#(2(x1))	(55)
5^#(4(1(4(x1))))	→	5^#(3(4(4(x1))))	(56)
0^#(1(2(0(4(x1)))))	→	0^#(2(3(1(0(x1)))))	(57)
0^#(1(2(4(x1))))	→	0^#(4(x1))	(58)
5^#(4(3(2(0(x1)))))	→	0^#(3(2(3(x1))))	(59)
0^#(4(2(0(x1))))	→	0^#(2(3(0(x1))))	(60)
0^#(1(2(4(x1))))	→	0^#(2(3(4(1(x1)))))	(61)
5^#(1(2(0(x1))))	→	5^#(x1)	(62)
0^#(1(1(2(x1))))	→	0^#(1(2(x1)))	(63)
5^#(1(2(4(0(x1)))))	→	5^#(0(3(2(1(4(x1))))))	(64)
0^#(1(2(0(x1))))	→	0^#(2(2(x1)))	(65)
0^#(1(2(0(x1))))	→	0^#(0(2(2(x1))))	(66)
0^#(4(2(0(x1))))	→	5^#(2(x1))	(67)
0^#(4(2(4(x1))))	→	0^#(4(2(0(3(x1)))))	(68)
5^#(1(2(4(x1))))	→	5^#(5(2(1(4(x1)))))	(69)
5^#(4(2(0(x1))))	→	5^#(x1)	(70)
5^#(4(1(4(x1))))	→	5^#(1(3(4(5(4(x1))))))	(71)
5^#(1(4(2(x1))))	→	5^#(2(1(x1)))	(72)
5^#(1(2(0(x1))))	→	0^#(2(5(x1)))	(73)
5^#(1(2(0(x1))))	→	5^#(0(x1))	(74)
5^#(1(2(0(x1))))	→	0^#(0(1(5(x1))))	(75)
0^#(1(2(0(4(x1)))))	→	0^#(x1)	(52)
5^#(1(2(2(4(x1)))))	→	5^#(x1)	(76)
5^#(0(1(2(4(x1)))))	→	5^#(x1)	(77)
0^#(4(2(0(x1))))	→	0^#(5(2(x1)))	(78)
5^#(4(5(2(0(x1)))))	→	5^#(0(5(4(4(2(x1))))))	(79)
0^#(1(2(4(x1))))	→	0^#(x1)	(80)
0^#(1(1(2(x1))))	→	0^#(1(x1))	(81)
0^#(1(2(4(x1))))	→	5^#(x1)	(82)
5^#(4(5(2(0(x1)))))	→	5^#(4(4(2(x1))))	(83)
5^#(4(1(4(x1))))	→	5^#(1(5(3(4(4(x1))))))	(84)
0^#(4(2(0(x1))))	→	0^#(4(0(x1)))	(85)
0^#(1(2(0(4(x1)))))	→	0^#(2(4(1(3(0(x1))))))	(86)
5^#(4(1(4(x1))))	→	5^#(0(4(x1)))	(87)
0^#(4(2(4(x1))))	→	0^#(2(3(1(4(x1)))))	(88)
0^#(4(2(0(x1))))	→	0^#(0(5(2(x1))))	(89)
5^#(4(1(1(4(x1)))))	→	5^#(4(x1))	(90)
0^#(4(2(4(x1))))	→	0^#(3(x1))	(91)
0^#(1(2(4(x1))))	→	0^#(2(3(1(x1))))	(92)
5^#(1(2(0(x1))))	→	0^#(1(5(x1)))	(93)
5^#(1(2(4(0(x1)))))	→	0^#(3(2(1(4(x1)))))	(94)
5^#(1(2(4(x1))))	→	5^#(2(1(3(4(x1)))))	(95)
0^#(1(4(2(2(x1)))))	→	0^#(2(x1))	(96)
5^#(1(4(1(2(x1)))))	→	5^#(3(4(1(x1))))	(97)
5^#(1(2(4(x1))))	→	5^#(2(1(4(x1))))	(98)
5^#(0(1(2(4(x1)))))	→	0^#(2(1(4(5(x1)))))	(99)
5^#(1(2(4(x1))))	→	5^#(2(1(x1)))	(100)
0^#(4(2(0(x1))))	→	0^#(4(0(0(x1))))	(101)
0^#(1(2(4(x1))))	→	0^#(3(x1))	(102)
5^#(4(2(0(x1))))	→	0^#(1(2(3(4(5(x1))))))	(103)
0^#(1(2(4(x1))))	→	0^#(3(4(x1)))	(104)
5^#(4(2(0(x1))))	→	5^#(3(3(0(2(x1)))))	(105)
5^#(1(3(1(4(x1)))))	→	0^#(x1)	(106)
0^#(1(2(4(x1))))	→	0^#(5(x1))	(107)
0^#(4(2(0(x1))))	→	0^#(0(x1))	(108)
5^#(1(3(1(4(x1)))))	→	5^#(1(3(4(0(x1)))))	(109)
5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
0^#(1(2(0(4(x1)))))	→	0^#(3(4(0(1(x1)))))	(110)
5^#(4(1(4(x1))))	→	0^#(4(x1))	(111)
0^#(1(2(4(x1))))	→	0^#(4(2(3(x1))))	(112)
5^#(1(2(4(x1))))	→	0^#(3(4(5(2(1(x1))))))	(113)
5^#(1(2(4(x1))))	→	5^#(1(4(x1)))	(114)
5^#(1(2(0(x1))))	→	5^#(x1)	(62)
5^#(4(3(2(0(x1)))))	→	5^#(4(0(3(2(3(x1))))))	(115)
0^#(4(2(0(x1))))	→	0^#(4(2(3(0(x1)))))	(116)
5^#(4(1(4(0(x1)))))	→	5^#(1(x1))	(117)
5^#(1(2(4(0(x1)))))	→	0^#(2(1(3(4(x1)))))	(118)
5^#(4(5(2(0(x1)))))	→	0^#(5(4(4(2(x1)))))	(119)
0^#(1(1(2(x1))))	→	0^#(2(x1))	(120)
5^#(4(2(0(x1))))	→	5^#(4(2(3(0(x1)))))	(121)
5^#(1(2(2(4(x1)))))	→	5^#(2(2(1(4(5(x1))))))	(122)
5^#(4(1(4(0(x1)))))	→	0^#(4(5(1(x1))))	(123)
5^#(1(4(2(x1))))	→	5^#(x1)	(124)
5^#(1(4(2(x1))))	→	5^#(4(x1))	(125)
0^#(1(2(0(4(x1)))))	→	0^#(1(x1))	(126)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(2(0(4(x1)))))	→	0^#(1(x1))	(126)
5^#(1(4(2(x1))))	→	5^#(4(x1))	(125)
5^#(1(2(0(x1))))	→	0^#(1(5(x1)))	(93)
5^#(1(4(2(x1))))	→	5^#(x1)	(124)
5^#(4(1(1(4(x1)))))	→	5^#(4(x1))	(90)
5^#(4(1(4(0(x1)))))	→	0^#(4(5(1(x1))))	(123)
5^#(4(1(4(x1))))	→	5^#(0(4(x1)))	(87)
0^#(4(2(0(x1))))	→	0^#(4(0(x1)))	(85)
0^#(1(2(4(x1))))	→	5^#(x1)	(82)
0^#(1(1(2(x1))))	→	0^#(1(x1))	(81)
0^#(1(2(4(x1))))	→	0^#(x1)	(80)
5^#(4(1(4(0(x1)))))	→	5^#(1(x1))	(117)
5^#(1(2(0(x1))))	→	5^#(x1)	(62)
5^#(1(2(4(x1))))	→	5^#(1(4(x1)))	(114)
5^#(0(1(2(4(x1)))))	→	5^#(x1)	(77)
5^#(1(2(2(4(x1)))))	→	5^#(x1)	(76)
0^#(1(2(0(4(x1)))))	→	0^#(x1)	(52)
5^#(1(2(0(x1))))	→	0^#(0(1(5(x1))))	(75)
5^#(4(1(4(x1))))	→	0^#(4(x1))	(111)
5^#(1(2(0(x1))))	→	5^#(0(x1))	(74)
5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
5^#(4(2(0(x1))))	→	5^#(x1)	(70)
0^#(4(2(4(x1))))	→	0^#(4(2(0(3(x1)))))	(68)
0^#(4(2(0(x1))))	→	0^#(0(x1))	(108)
0^#(1(1(2(x1))))	→	0^#(1(2(x1)))	(63)
0^#(1(2(4(x1))))	→	0^#(5(x1))	(107)
5^#(1(2(0(x1))))	→	5^#(x1)	(62)
5^#(1(3(1(4(x1)))))	→	0^#(x1)	(106)
0^#(4(2(0(x1))))	→	0^#(4(0(0(x1))))	(101)
0^#(1(2(4(x1))))	→	0^#(4(x1))	(58)
5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
0^#(1(2(0(4(x1)))))	→	0^#(x1)	(52)

1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

0	1
0	0

· x₁ +

[1(x₁)]

0	0
0	1

· x₁ +

[4(x₁)]

0	1
0	1

· x₁ +

45799

[5(x₁)]

0	1
0	1

· x₁ +

45799

[3(x₁)]

0	0
0	1

· x₁ +

[0(x₁)]

1	1
0	1

· x₁ +

45799

[5^#(x₁)]

0	1
0	0

· x₁ +

[2(x₁)]

0	0
1	1

· x₁ +

15307

30492

together with the usable rules

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

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

0^#(1(2(0(4(x1)))))	→	0^#(1(x1))	(126)
5^#(1(4(2(x1))))	→	5^#(4(x1))	(125)
5^#(1(2(0(x1))))	→	0^#(1(5(x1)))	(93)
5^#(1(4(2(x1))))	→	5^#(x1)	(124)
5^#(4(1(4(0(x1)))))	→	0^#(4(5(1(x1))))	(123)
0^#(4(2(0(x1))))	→	0^#(4(0(x1)))	(85)
0^#(1(2(4(x1))))	→	5^#(x1)	(82)
0^#(1(1(2(x1))))	→	0^#(1(x1))	(81)
0^#(1(2(4(x1))))	→	0^#(x1)	(80)
5^#(1(2(0(x1))))	→	5^#(x1)	(62)
5^#(1(2(4(x1))))	→	5^#(1(4(x1)))	(114)
5^#(0(1(2(4(x1)))))	→	5^#(x1)	(77)
5^#(1(2(2(4(x1)))))	→	5^#(x1)	(76)
0^#(1(2(0(4(x1)))))	→	0^#(x1)	(52)
5^#(1(2(0(x1))))	→	0^#(0(1(5(x1))))	(75)
5^#(4(1(4(x1))))	→	0^#(4(x1))	(111)
5^#(1(2(0(x1))))	→	5^#(0(x1))	(74)
5^#(4(2(0(x1))))	→	5^#(x1)	(70)
0^#(4(2(0(x1))))	→	0^#(0(x1))	(108)
0^#(1(2(4(x1))))	→	0^#(5(x1))	(107)
5^#(1(2(0(x1))))	→	5^#(x1)	(62)
5^#(1(3(1(4(x1)))))	→	0^#(x1)	(106)
0^#(4(2(0(x1))))	→	0^#(4(0(0(x1))))	(101)
0^#(1(2(4(x1))))	→	0^#(4(x1))	(58)
0^#(1(2(0(4(x1)))))	→	0^#(x1)	(52)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
5^#(4(1(4(x1))))	→	5^#(0(4(x1)))	(87)
5^#(4(1(1(4(x1)))))	→	5^#(4(x1))	(90)

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	1
1	1

· x₁ +

[4(x₁)]

0	1
0	1

· x₁ +

[5(x₁)]

x₁ +

[3(x₁)]

[0(x₁)]

x₁ +

[5^#(x₁)]

1	0
0	0

· x₁ +

[2(x₁)]

0	1
1	0

· x₁ +

together with the usable rules

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

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

5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
5^#(4(1(4(x1))))	→	5^#(4(x1))	(54)
5^#(4(1(4(x1))))	→	5^#(0(4(x1)))	(87)
5^#(4(1(1(4(x1)))))	→	5^#(4(x1))	(90)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.