Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212117)

The rewrite relation of the following TRS is considered.

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

There are 124 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^#(1(4(x1)))	→	1^#(x1)	(162)
0^#(1(4(5(x1))))	→	5^#(1(x1))	(115)
0^#(1(4(x1)))	→	0^#(1(x1))	(116)
5^#(2(5(1(x1))))	→	5^#(1(5(2(x1))))	(159)
0^#(1(4(x1)))	→	0^#(x1)	(74)
1^#(1(2(4(x1))))	→	1^#(2(x1))	(158)
1^#(2(4(x1)))	→	5^#(1(x1))	(113)
0^#(1(1(5(4(x1)))))	→	1^#(1(x1))	(110)
1^#(0(4(2(1(x1)))))	→	1^#(0(x1))	(156)
1^#(0(1(4(x1))))	→	1^#(2(1(x1)))	(105)
0^#(1(4(5(x1))))	→	0^#(1(x1))	(102)
0^#(1(4(x1)))	→	0^#(x1)	(74)
1^#(2(5(0(1(x1)))))	→	5^#(1(0(1(x1))))	(98)
0^#(1(1(x1)))	→	0^#(x1)	(149)
1^#(2(5(0(1(x1)))))	→	1^#(0(1(x1)))	(96)
1^#(0(4(2(1(x1)))))	→	0^#(x1)	(148)
0^#(1(4(x1)))	→	0^#(x1)	(74)
0^#(1(4(x1)))	→	0^#(x1)	(74)
5^#(2(4(x1)))	→	5^#(2(x1))	(143)
1^#(0(1(4(x1))))	→	0^#(1(2(1(x1))))	(90)
0^#(1(1(5(x1))))	→	5^#(1(2(x1)))	(89)
1^#(0(1(4(x1))))	→	1^#(x1)	(88)
5^#(2(4(x1)))	→	0^#(5(x1))	(142)
1^#(2(4(x1)))	→	1^#(x1)	(86)
0^#(1(4(5(x1))))	→	0^#(5(1(x1)))	(87)
5^#(0(1(4(x1))))	→	1^#(x1)	(141)
0^#(5(1(4(x1))))	→	1^#(x1)	(85)
5^#(0(1(4(x1))))	→	0^#(5(3(1(x1))))	(139)
0^#(1(4(x1)))	→	5^#(x1)	(84)
1^#(2(4(x1)))	→	1^#(2(x1))	(82)
5^#(0(5(2(4(x1)))))	→	5^#(5(2(x1)))	(137)
5^#(0(5(2(4(x1)))))	→	0^#(5(5(2(x1))))	(80)
5^#(2(5(1(x1))))	→	5^#(2(x1))	(79)
0^#(1(4(x1)))	→	0^#(x1)	(74)
5^#(0(5(2(4(x1)))))	→	5^#(2(x1))	(77)
0^#(1(4(x1)))	→	5^#(0(5(x1)))	(78)
5^#(0(5(2(4(x1)))))	→	0^#(0(5(5(2(x1)))))	(75)
0^#(1(4(x1)))	→	0^#(x1)	(74)
1^#(2(4(x1)))	→	1^#(x1)	(86)
0^#(1(4(x1)))	→	5^#(0(1(x1)))	(70)
1^#(5(2(1(x1))))	→	5^#(1(2(x1)))	(69)
1^#(2(2(4(x1))))	→	1^#(x1)	(133)
0^#(1(4(5(x1))))	→	1^#(x1)	(65)
0^#(1(4(x1)))	→	5^#(5(0(1(x1))))	(68)
5^#(2(5(1(x1))))	→	1^#(5(2(x1)))	(130)
0^#(1(4(5(x1))))	→	1^#(x1)	(65)
1^#(5(2(1(x1))))	→	1^#(2(x1))	(128)
5^#(2(4(x1)))	→	5^#(x1)	(53)
5^#(2(4(x1)))	→	5^#(5(x1))	(59)
0^#(1(1(5(x1))))	→	1^#(2(x1))	(123)
1^#(0(0(2(4(x1)))))	→	1^#(x1)	(58)
0^#(1(1(5(4(x1)))))	→	1^#(x1)	(56)
0^#(1(4(x1)))	→	0^#(5(x1))	(122)
5^#(2(4(x1)))	→	5^#(x1)	(53)

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
1	1

· x₁ +

[4(x₁)]

x₁ +

7314

30774

[5(x₁)]

x₁ +

[3(x₁)]

[0(x₁)]

1	0
1	0

· x₁ +

[5^#(x₁)]

1	0
0	0

· x₁ +

[2(x₁)]

1	1
0	0

· x₁ +

[1^#(x₁)]

1	1
0	0

· x₁ +

together with the usable rules

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

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

0^#(1(4(x1)))	→	1^#(x1)	(162)
0^#(1(4(5(x1))))	→	5^#(1(x1))	(115)
0^#(1(4(x1)))	→	0^#(1(x1))	(116)
0^#(1(4(x1)))	→	0^#(x1)	(74)
1^#(1(2(4(x1))))	→	1^#(2(x1))	(158)
1^#(2(4(x1)))	→	5^#(1(x1))	(113)
0^#(1(1(5(4(x1)))))	→	1^#(1(x1))	(110)
1^#(0(4(2(1(x1)))))	→	1^#(0(x1))	(156)
1^#(0(1(4(x1))))	→	1^#(2(1(x1)))	(105)
0^#(1(4(5(x1))))	→	0^#(1(x1))	(102)
0^#(1(4(x1)))	→	0^#(x1)	(74)
1^#(0(4(2(1(x1)))))	→	0^#(x1)	(148)
0^#(1(4(x1)))	→	0^#(x1)	(74)
0^#(1(4(x1)))	→	0^#(x1)	(74)
5^#(2(4(x1)))	→	5^#(2(x1))	(143)
1^#(0(1(4(x1))))	→	0^#(1(2(1(x1))))	(90)
0^#(1(1(5(x1))))	→	5^#(1(2(x1)))	(89)
1^#(0(1(4(x1))))	→	1^#(x1)	(88)
5^#(2(4(x1)))	→	0^#(5(x1))	(142)
1^#(2(4(x1)))	→	1^#(x1)	(86)
0^#(1(4(5(x1))))	→	0^#(5(1(x1)))	(87)
5^#(0(1(4(x1))))	→	1^#(x1)	(141)
0^#(5(1(4(x1))))	→	1^#(x1)	(85)
5^#(0(1(4(x1))))	→	0^#(5(3(1(x1))))	(139)
0^#(1(4(x1)))	→	5^#(x1)	(84)
1^#(2(4(x1)))	→	1^#(2(x1))	(82)
5^#(0(5(2(4(x1)))))	→	5^#(5(2(x1)))	(137)
5^#(0(5(2(4(x1)))))	→	0^#(5(5(2(x1))))	(80)
0^#(1(4(x1)))	→	0^#(x1)	(74)
5^#(0(5(2(4(x1)))))	→	5^#(2(x1))	(77)
0^#(1(4(x1)))	→	5^#(0(5(x1)))	(78)
5^#(0(5(2(4(x1)))))	→	0^#(0(5(5(2(x1)))))	(75)
0^#(1(4(x1)))	→	0^#(x1)	(74)
1^#(2(4(x1)))	→	1^#(x1)	(86)
0^#(1(4(x1)))	→	5^#(0(1(x1)))	(70)
1^#(2(2(4(x1))))	→	1^#(x1)	(133)
0^#(1(4(5(x1))))	→	1^#(x1)	(65)
0^#(1(4(x1)))	→	5^#(5(0(1(x1))))	(68)
0^#(1(4(5(x1))))	→	1^#(x1)	(65)
5^#(2(4(x1)))	→	5^#(x1)	(53)
5^#(2(4(x1)))	→	5^#(5(x1))	(59)
0^#(1(1(5(x1))))	→	1^#(2(x1))	(123)
1^#(0(0(2(4(x1)))))	→	1^#(x1)	(58)
0^#(1(1(5(4(x1)))))	→	1^#(x1)	(56)
0^#(1(4(x1)))	→	0^#(5(x1))	(122)
5^#(2(4(x1)))	→	5^#(x1)	(53)

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

1^#(5(2(1(x1))))	→	1^#(2(x1))	(128)
1^#(5(2(1(x1))))	→	5^#(1(2(x1)))	(69)
1^#(2(5(0(1(x1)))))	→	1^#(0(1(x1)))	(96)
1^#(2(5(0(1(x1)))))	→	5^#(1(0(1(x1))))	(98)
5^#(2(5(1(x1))))	→	5^#(2(x1))	(79)
5^#(2(5(1(x1))))	→	1^#(5(2(x1)))	(130)
5^#(2(5(1(x1))))	→	5^#(1(5(2(x1))))	(159)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

1^#(5(2(1(x1))))	→	1^#(2(x1))	(128)
1^#(5(2(1(x1))))	→	5^#(1(2(x1)))	(69)
1^#(2(5(0(1(x1)))))	→	1^#(0(1(x1)))	(96)
1^#(2(5(0(1(x1)))))	→	5^#(1(0(1(x1))))	(98)
5^#(2(5(1(x1))))	→	5^#(2(x1))	(79)
5^#(2(5(1(x1))))	→	1^#(5(2(x1)))	(130)

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

→

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

(159)

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)]

0	1
0	1

· x₁ +

21071

[4(x₁)]

21071

[5(x₁)]

0	0
1	1

· x₁ +

21072

[3(x₁)]

1	0
0	0

· x₁ +

[0(x₁)]

21072

42142

[5^#(x₁)]

1	0
0	0

· x₁ +

[2(x₁)]

0	1
0	1

· x₁ +

[1^#(x₁)]

together with the usable rules

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

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

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

→

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

(159)

could be deleted.

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

0^#(1(1(x1)))

→

0^#(x1)

(149)

1.1.1.1.2 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)	=	2	status(1)	=	[1]	list-extension(1)	=	Lex
prec(4)	=	0	status(4)	=	[]	list-extension(4)	=	Lex
prec(5)	=	2	status(5)	=	[1]	list-extension(5)	=	Lex
prec(3)	=	1	status(3)	=	[]	list-extension(3)	=	Lex
prec(0)	=	2	status(0)	=	[1]	list-extension(0)	=	Lex
prec(5^#)	=	0	status(5^#)	=	[]	list-extension(5^#)	=	Lex
prec(2)	=	1	status(2)	=	[]	list-extension(2)	=	Lex
prec(1^#)	=	1	status(1^#)	=	[]	list-extension(1^#)	=	Lex

and the following Max-polynomial interpretation

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

together with the usable rules

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

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

0^#(1(1(x1)))

→

0^#(x1)

(149)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.