Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212037)

The rewrite relation of the following TRS is considered.

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

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 134 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^#(5(3(4(1(x1)))))	→	2^#(x1)	(179)
0^#(1(5(4(2(x1)))))	→	4^#(3(x1))	(178)
0^#(1(4(x1)))	→	0^#(0(x1))	(122)
2^#(4(3(2(x1))))	→	2^#(0(4(2(3(x1)))))	(176)
0^#(1(5(4(2(x1)))))	→	0^#(4(3(x1)))	(120)
2^#(4(2(0(1(x1)))))	→	4^#(x1)	(118)
2^#(4(3(2(x1))))	→	4^#(2(3(x1)))	(116)
2^#(0(3(1(x1))))	→	0^#(3(2(1(1(x1)))))	(114)
0^#(1(5(3(2(x1)))))	→	0^#(5(0(3(2(1(x1))))))	(169)
0^#(3(2(x1)))	→	0^#(2(3(x1)))	(102)
0^#(3(2(4(x1))))	→	0^#(4(2(3(0(x1)))))	(168)
0^#(3(2(x1)))	→	0^#(0(2(3(x1))))	(166)
0^#(5(4(1(4(x1)))))	→	0^#(x1)	(95)
2^#(5(3(4(1(x1)))))	→	0^#(4(3(2(x1))))	(164)
0^#(2(2(4(3(x1)))))	→	2^#(x1)	(92)
0^#(1(4(x1)))	→	0^#(x1)	(161)
2^#(4(1(3(x1))))	→	0^#(4(3(2(1(x1)))))	(91)
2^#(4(1(5(3(x1)))))	→	2^#(x1)	(160)
4^#(2(0(3(x1))))	→	0^#(4(3(x1)))	(158)
2^#(4(3(2(x1))))	→	0^#(4(2(3(x1))))	(155)
0^#(3(2(4(3(x1)))))	→	2^#(x1)	(89)
0^#(3(4(x1)))	→	0^#(0(4(3(x1))))	(154)
0^#(3(2(4(x1))))	→	4^#(2(3(0(x1))))	(153)
0^#(3(0(4(5(x1)))))	→	0^#(0(4(3(2(x1)))))	(84)
2^#(5(3(4(1(x1)))))	→	4^#(3(2(x1)))	(150)
0^#(1(5(3(2(x1)))))	→	0^#(3(2(1(x1))))	(81)
4^#(3(4(3(x1))))	→	4^#(3(0(4(3(x1)))))	(77)
0^#(2(2(4(3(x1)))))	→	4^#(3(0(2(x1))))	(71)
0^#(2(2(4(3(x1)))))	→	0^#(2(x1))	(69)
0^#(3(2(4(x1))))	→	0^#(x1)	(143)
2^#(4(2(0(1(x1)))))	→	2^#(0(4(x1)))	(141)
0^#(3(4(x1)))	→	4^#(3(x1))	(140)
0^#(2(2(4(3(x1)))))	→	0^#(4(3(0(2(x1)))))	(63)
0^#(3(0(4(5(x1)))))	→	0^#(4(3(2(x1))))	(62)
0^#(2(2(4(3(x1)))))	→	2^#(0(4(3(0(2(x1))))))	(136)
0^#(3(4(x1)))	→	0^#(4(3(x1)))	(134)
0^#(3(0(4(5(x1)))))	→	2^#(x1)	(133)
0^#(3(0(4(5(x1)))))	→	4^#(3(2(x1)))	(131)
0^#(3(2(4(3(x1)))))	→	4^#(2(x1))	(130)
4^#(3(4(3(x1))))	→	0^#(4(3(x1)))	(128)
2^#(4(2(0(1(x1)))))	→	0^#(4(x1))	(49)
4^#(2(0(3(x1))))	→	4^#(3(x1))	(47)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

2^#(5(3(4(1(x1)))))	→	2^#(x1)	(179)
0^#(1(5(4(2(x1)))))	→	4^#(3(x1))	(178)
0^#(1(5(4(2(x1)))))	→	0^#(4(3(x1)))	(120)
0^#(5(4(1(4(x1)))))	→	0^#(x1)	(95)
2^#(5(3(4(1(x1)))))	→	0^#(4(3(2(x1))))	(164)
2^#(4(1(5(3(x1)))))	→	2^#(x1)	(160)
0^#(3(0(4(5(x1)))))	→	0^#(0(4(3(2(x1)))))	(84)
2^#(5(3(4(1(x1)))))	→	4^#(3(2(x1)))	(150)
0^#(1(5(3(2(x1)))))	→	0^#(3(2(1(x1))))	(81)
0^#(3(0(4(5(x1)))))	→	0^#(4(3(2(x1))))	(62)
0^#(3(0(4(5(x1)))))	→	2^#(x1)	(133)
0^#(3(0(4(5(x1)))))	→	4^#(3(2(x1)))	(131)

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

0^#(1(4(x1)))	→	0^#(x1)	(161)
0^#(1(4(x1)))	→	0^#(0(x1))	(122)
0^#(3(2(4(x1))))	→	0^#(x1)	(143)
0^#(3(2(4(x1))))	→	4^#(2(3(0(x1))))	(153)
0^#(3(2(4(x1))))	→	0^#(4(2(3(0(x1)))))	(168)
2^#(0(3(1(x1))))	→	0^#(3(2(1(1(x1)))))	(114)
2^#(4(3(2(x1))))	→	4^#(2(3(x1)))	(116)
2^#(4(3(2(x1))))	→	0^#(4(2(3(x1))))	(155)
2^#(4(3(2(x1))))	→	2^#(0(4(2(3(x1)))))	(176)
4^#(2(0(3(x1))))	→	4^#(3(x1))	(47)
4^#(2(0(3(x1))))	→	0^#(4(3(x1)))	(158)
0^#(3(2(x1)))	→	0^#(2(3(x1)))	(102)
0^#(3(2(x1)))	→	0^#(0(2(3(x1))))	(166)
0^#(2(2(4(3(x1)))))	→	2^#(x1)	(92)
0^#(2(2(4(3(x1)))))	→	0^#(2(x1))	(69)
0^#(2(2(4(3(x1)))))	→	4^#(3(0(2(x1))))	(71)
0^#(2(2(4(3(x1)))))	→	0^#(4(3(0(2(x1)))))	(63)
0^#(2(2(4(3(x1)))))	→	2^#(0(4(3(0(2(x1))))))	(136)
0^#(3(4(x1)))	→	4^#(3(x1))	(140)
0^#(3(4(x1)))	→	0^#(4(3(x1)))	(134)
0^#(3(4(x1)))	→	0^#(0(4(3(x1))))	(154)
2^#(4(1(3(x1))))	→	0^#(4(3(2(1(x1)))))	(91)
2^#(4(2(0(1(x1)))))	→	4^#(x1)	(118)
2^#(4(2(0(1(x1)))))	→	0^#(4(x1))	(49)
2^#(4(2(0(1(x1)))))	→	2^#(0(4(x1)))	(141)
0^#(3(2(4(3(x1)))))	→	2^#(x1)	(89)
0^#(3(2(4(3(x1)))))	→	4^#(2(x1))	(130)
4^#(3(4(3(x1))))	→	0^#(4(3(x1)))	(128)
4^#(3(4(3(x1))))	→	4^#(3(0(4(3(x1)))))	(77)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

[1(x₁)]

1	1
0	0

· x₁ +

[4(x₁)]

0	1
1	1

· x₁ +

[5(x₁)]

0	1
0	1

· x₁ +

[3(x₁)]

0	1
0	1

· x₁ +

[2^#(x₁)]

0	1
0	0

· x₁ +

[4^#(x₁)]

1	0
0	0

· x₁ +

[0(x₁)]

1	0
1	0

· x₁ +

[2(x₁)]

0	1
0	1

· x₁ +

together with the usable rules

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

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

0^#(1(4(x1)))	→	0^#(x1)	(161)
0^#(1(4(x1)))	→	0^#(0(x1))	(122)
0^#(3(2(4(x1))))	→	0^#(x1)	(143)
0^#(3(2(4(x1))))	→	4^#(2(3(0(x1))))	(153)
0^#(3(2(4(x1))))	→	0^#(4(2(3(0(x1)))))	(168)
2^#(0(3(1(x1))))	→	0^#(3(2(1(1(x1)))))	(114)
2^#(4(3(2(x1))))	→	4^#(2(3(x1)))	(116)
2^#(4(3(2(x1))))	→	0^#(4(2(3(x1))))	(155)
2^#(4(3(2(x1))))	→	2^#(0(4(2(3(x1)))))	(176)
4^#(2(0(3(x1))))	→	4^#(3(x1))	(47)
4^#(2(0(3(x1))))	→	0^#(4(3(x1)))	(158)
0^#(2(2(4(3(x1)))))	→	2^#(x1)	(92)
0^#(2(2(4(3(x1)))))	→	0^#(2(x1))	(69)
0^#(2(2(4(3(x1)))))	→	4^#(3(0(2(x1))))	(71)
0^#(2(2(4(3(x1)))))	→	0^#(4(3(0(2(x1)))))	(63)
0^#(3(4(x1)))	→	4^#(3(x1))	(140)
0^#(3(4(x1)))	→	0^#(4(3(x1)))	(134)
0^#(3(4(x1)))	→	0^#(0(4(3(x1))))	(154)
2^#(4(1(3(x1))))	→	0^#(4(3(2(1(x1)))))	(91)
2^#(4(2(0(1(x1)))))	→	4^#(x1)	(118)
2^#(4(2(0(1(x1)))))	→	0^#(4(x1))	(49)
2^#(4(2(0(1(x1)))))	→	2^#(0(4(x1)))	(141)
0^#(3(2(4(3(x1)))))	→	2^#(x1)	(89)
0^#(3(2(4(3(x1)))))	→	4^#(2(x1))	(130)
4^#(3(4(3(x1))))	→	0^#(4(3(x1)))	(128)
4^#(3(4(3(x1))))	→	4^#(3(0(4(3(x1)))))	(77)

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

0^#(3(2(x1)))	→	0^#(2(3(x1)))	(102)
0^#(3(2(x1)))	→	0^#(0(2(3(x1))))	(166)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	1
[4(x₁)]	=	1
[5(x₁)]	=	1
[3(x₁)]	=	x₁ + 436
[2^#(x₁)]	=	0
[4^#(x₁)]	=	0
[0(x₁)]	=	1
[2(x₁)]	=	1

together with the usable rules

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

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

0^#(3(2(x1)))	→	0^#(2(3(x1)))	(102)
0^#(3(2(x1)))	→	0^#(0(2(3(x1))))	(166)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.