Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212892)

The rewrite relation of the following TRS is considered.

0(0(x1))	→	1(0(2(0(2(x1)))))	(1)
0(0(x1))	→	0(2(3(4(0(2(x1))))))	(2)
0(3(x1))	→	0(2(3(2(x1))))	(3)
0(3(x1))	→	2(0(2(1(3(x1)))))	(4)
0(3(x1))	→	2(3(0(2(2(x1)))))	(5)
0(3(x1))	→	0(2(2(2(3(2(x1))))))	(6)
0(0(0(x1)))	→	0(2(0(0(x1))))	(7)
0(0(4(x1)))	→	2(4(0(0(3(2(x1))))))	(8)
0(0(5(x1)))	→	0(5(2(0(x1))))	(9)
0(1(3(x1)))	→	3(2(2(1(0(x1)))))	(10)
0(3(0(x1)))	→	2(0(2(3(4(0(x1))))))	(11)
0(4(3(x1)))	→	3(0(2(4(x1))))	(12)
0(4(3(x1)))	→	1(0(2(1(3(4(x1))))))	(13)
0(4(3(x1)))	→	2(2(4(3(4(0(x1))))))	(14)
0(4(5(x1)))	→	2(4(0(5(2(5(x1))))))	(15)
0(4(5(x1)))	→	5(3(2(4(4(0(x1))))))	(16)
0(5(0(x1)))	→	0(5(2(4(0(x1)))))	(17)
0(5(0(x1)))	→	5(0(2(3(2(0(x1))))))	(18)
0(5(3(x1)))	→	3(2(5(3(2(0(x1))))))	(19)
1(0(0(x1)))	→	1(0(2(5(0(2(x1))))))	(20)
1(0(3(x1)))	→	2(1(3(0(5(2(x1))))))	(21)
1(0(4(x1)))	→	3(0(2(1(4(x1)))))	(22)
1(0(4(x1)))	→	2(1(2(1(4(0(x1))))))	(23)
3(0(0(x1)))	→	3(0(2(0(x1))))	(24)
3(0(3(x1)))	→	3(0(2(3(x1))))	(25)
3(0(3(x1)))	→	3(3(2(2(0(x1)))))	(26)
3(0(4(x1)))	→	2(2(3(4(0(x1)))))	(27)
3(0(4(x1)))	→	2(3(4(2(0(3(x1))))))	(28)
3(0(4(x1)))	→	5(5(2(4(0(3(x1))))))	(29)
3(3(4(x1)))	→	3(2(1(3(2(4(x1))))))	(30)
3(5(0(x1)))	→	3(5(1(0(2(1(x1))))))	(31)
4(0(4(x1)))	→	0(2(4(4(0(x1)))))	(32)
4(0(4(x1)))	→	1(4(0(2(4(0(x1))))))	(33)
0(0(4(3(x1))))	→	3(0(0(2(1(4(x1))))))	(34)
0(0(5(3(x1))))	→	0(5(2(5(0(3(x1))))))	(35)
0(1(0(3(x1))))	→	2(1(0(3(0(x1)))))	(36)
0(1(1(0(x1))))	→	2(1(0(0(1(2(x1))))))	(37)
0(4(0(0(x1))))	→	0(4(0(2(1(0(x1))))))	(38)
0(4(1(3(x1))))	→	2(1(4(2(0(3(x1))))))	(39)
0(4(5(3(x1))))	→	4(3(2(5(0(x1)))))	(40)
0(4(5(3(x1))))	→	4(3(5(0(2(4(x1))))))	(41)
0(5(0(4(x1))))	→	5(0(0(2(4(x1)))))	(42)
0(5(5(3(x1))))	→	2(3(5(5(0(x1)))))	(43)
1(3(0(4(x1))))	→	4(0(1(2(3(x1)))))	(44)
4(0(3(5(x1))))	→	5(2(3(4(2(0(x1))))))	(45)
0(0(1(0(4(x1)))))	→	0(0(0(1(4(2(x1))))))	(46)
0(0(5(5(3(x1)))))	→	0(5(0(3(2(5(x1))))))	(47)
3(0(4(3(4(x1)))))	→	3(0(3(4(2(4(x1))))))	(48)
3(3(0(4(5(x1)))))	→	1(4(5(3(0(3(x1))))))	(49)
3(4(2(0(3(x1)))))	→	2(3(0(2(3(4(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 160 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

3^#(3(0(4(5(x1)))))	→	3^#(x1)	(201)
3^#(0(3(x1)))	→	0^#(x1)	(135)
3^#(0(4(x1)))	→	3^#(4(2(0(3(x1)))))	(200)
0^#(3(0(x1)))	→	4^#(0(x1))	(132)
0^#(0(5(x1)))	→	0^#(x1)	(133)
3^#(0(4(3(4(x1)))))	→	3^#(0(3(4(2(4(x1))))))	(131)
1^#(3(0(4(x1))))	→	3^#(x1)	(199)
3^#(0(4(x1)))	→	0^#(3(x1))	(129)
0^#(0(4(3(x1))))	→	3^#(0(0(2(1(4(x1))))))	(127)
0^#(1(0(3(x1))))	→	3^#(0(x1))	(125)
0^#(4(1(3(x1))))	→	0^#(3(x1))	(124)
3^#(3(0(4(5(x1)))))	→	0^#(3(x1))	(122)
1^#(0(4(x1)))	→	1^#(4(0(x1)))	(123)
3^#(0(4(3(4(x1)))))	→	0^#(3(4(2(4(x1)))))	(121)
0^#(0(1(0(4(x1)))))	→	0^#(0(0(1(4(2(x1))))))	(120)
0^#(4(5(x1)))	→	0^#(x1)	(194)
0^#(4(3(x1)))	→	4^#(3(4(0(x1))))	(118)
3^#(0(4(x1)))	→	0^#(3(x1))	(129)
4^#(0(4(x1)))	→	4^#(0(x1))	(76)
0^#(4(3(x1)))	→	1^#(3(4(x1)))	(117)
3^#(3(0(4(5(x1)))))	→	3^#(0(3(x1)))	(116)
4^#(0(4(x1)))	→	0^#(x1)	(67)
3^#(0(4(x1)))	→	0^#(x1)	(115)
0^#(4(5(x1)))	→	4^#(0(x1))	(190)
0^#(1(1(0(x1))))	→	1^#(0(0(1(2(x1)))))	(112)
0^#(4(3(x1)))	→	3^#(4(0(x1)))	(111)
0^#(1(0(3(x1))))	→	0^#(x1)	(110)
0^#(1(0(3(x1))))	→	1^#(0(3(0(x1))))	(182)
1^#(0(4(x1)))	→	4^#(0(x1))	(105)
3^#(5(0(x1)))	→	1^#(x1)	(181)
4^#(0(4(x1)))	→	4^#(4(0(x1)))	(180)
0^#(0(5(3(x1))))	→	0^#(3(x1))	(102)
3^#(0(4(x1)))	→	3^#(x1)	(99)
3^#(0(4(x1)))	→	3^#(x1)	(99)
0^#(3(0(x1)))	→	3^#(4(0(x1)))	(100)
3^#(0(4(3(4(x1)))))	→	3^#(4(2(4(x1))))	(171)
0^#(4(5(3(x1))))	→	4^#(3(5(0(2(4(x1))))))	(172)
0^#(1(0(3(x1))))	→	0^#(3(0(x1)))	(93)
3^#(0(4(x1)))	→	4^#(0(x1))	(169)
1^#(0(4(x1)))	→	1^#(4(x1))	(92)
4^#(0(3(5(x1))))	→	3^#(4(2(0(x1))))	(91)
0^#(5(5(3(x1))))	→	0^#(x1)	(90)
3^#(0(4(x1)))	→	3^#(4(0(x1)))	(88)
0^#(3(x1))	→	1^#(3(x1))	(87)
0^#(0(5(5(3(x1)))))	→	0^#(5(0(3(2(5(x1))))))	(163)
3^#(4(2(0(3(x1)))))	→	3^#(4(x1))	(162)
3^#(4(2(0(3(x1)))))	→	4^#(x1)	(83)
4^#(0(3(5(x1))))	→	0^#(x1)	(84)
1^#(0(4(x1)))	→	0^#(x1)	(80)
4^#(0(4(x1)))	→	4^#(0(x1))	(76)
0^#(0(4(x1)))	→	4^#(0(0(3(2(x1)))))	(73)
0^#(4(5(x1)))	→	4^#(4(0(x1)))	(74)
0^#(5(3(x1)))	→	0^#(x1)	(158)
0^#(4(3(x1)))	→	4^#(0(x1))	(72)
0^#(4(5(3(x1))))	→	4^#(x1)	(157)
0^#(0(4(x1)))	→	0^#(0(3(2(x1))))	(155)
0^#(0(5(5(3(x1)))))	→	0^#(3(2(5(x1))))	(156)
0^#(4(3(x1)))	→	4^#(x1)	(70)
0^#(1(3(x1)))	→	1^#(0(x1))	(68)
0^#(5(0(x1)))	→	4^#(0(x1))	(151)
4^#(0(4(x1)))	→	0^#(x1)	(67)
0^#(0(4(3(x1))))	→	1^#(4(x1))	(65)
0^#(4(3(x1)))	→	0^#(x1)	(150)
0^#(4(0(0(x1))))	→	1^#(0(x1))	(64)
0^#(4(3(x1)))	→	4^#(x1)	(70)
0^#(0(4(x1)))	→	0^#(3(2(x1)))	(61)
0^#(0(4(3(x1))))	→	4^#(x1)	(60)
0^#(1(3(x1)))	→	0^#(x1)	(146)
0^#(4(5(3(x1))))	→	0^#(x1)	(59)
0^#(4(3(x1)))	→	3^#(4(x1))	(58)
3^#(0(4(x1)))	→	4^#(0(3(x1)))	(141)
0^#(4(5(3(x1))))	→	3^#(5(0(2(4(x1)))))	(138)

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 +

13739

[1(x₁)]

x₁ +

[4(x₁)]

x₁ +

3329

[5(x₁)]

x₁ +

6012

[3(x₁)]

x₁ +

[4^#(x₁)]

0	1
0	0

· x₁ +

[0(x₁)]

1	1
1	1

· x₁ +

13738

[3^#(x₁)]

0	1
0	0

· x₁ +

[2(x₁)]

0	0
1	0

· x₁ +

[1^#(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

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

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

3^#(3(0(4(5(x1)))))	→	3^#(x1)	(201)
3^#(0(3(x1)))	→	0^#(x1)	(135)
3^#(0(4(x1)))	→	3^#(4(2(0(3(x1)))))	(200)
0^#(3(0(x1)))	→	4^#(0(x1))	(132)
0^#(0(5(x1)))	→	0^#(x1)	(133)
3^#(0(4(3(4(x1)))))	→	3^#(0(3(4(2(4(x1))))))	(131)
1^#(3(0(4(x1))))	→	3^#(x1)	(199)
3^#(0(4(x1)))	→	0^#(3(x1))	(129)
0^#(0(4(3(x1))))	→	3^#(0(0(2(1(4(x1))))))	(127)
0^#(1(0(3(x1))))	→	3^#(0(x1))	(125)
0^#(4(1(3(x1))))	→	0^#(3(x1))	(124)
3^#(3(0(4(5(x1)))))	→	0^#(3(x1))	(122)
3^#(0(4(3(4(x1)))))	→	0^#(3(4(2(4(x1)))))	(121)
0^#(4(5(x1)))	→	0^#(x1)	(194)
0^#(4(3(x1)))	→	4^#(3(4(0(x1))))	(118)
3^#(0(4(x1)))	→	0^#(3(x1))	(129)
4^#(0(4(x1)))	→	4^#(0(x1))	(76)
0^#(4(3(x1)))	→	1^#(3(4(x1)))	(117)
3^#(3(0(4(5(x1)))))	→	3^#(0(3(x1)))	(116)
4^#(0(4(x1)))	→	0^#(x1)	(67)
3^#(0(4(x1)))	→	0^#(x1)	(115)
0^#(4(5(x1)))	→	4^#(0(x1))	(190)
0^#(1(1(0(x1))))	→	1^#(0(0(1(2(x1)))))	(112)
0^#(4(3(x1)))	→	3^#(4(0(x1)))	(111)
0^#(1(0(3(x1))))	→	0^#(x1)	(110)
0^#(1(0(3(x1))))	→	1^#(0(3(0(x1))))	(182)
1^#(0(4(x1)))	→	4^#(0(x1))	(105)
3^#(5(0(x1)))	→	1^#(x1)	(181)
0^#(0(5(3(x1))))	→	0^#(3(x1))	(102)
3^#(0(4(x1)))	→	3^#(x1)	(99)
3^#(0(4(x1)))	→	3^#(x1)	(99)
0^#(3(0(x1)))	→	3^#(4(0(x1)))	(100)
3^#(0(4(3(4(x1)))))	→	3^#(4(2(4(x1))))	(171)
0^#(4(5(3(x1))))	→	4^#(3(5(0(2(4(x1))))))	(172)
0^#(1(0(3(x1))))	→	0^#(3(0(x1)))	(93)
3^#(0(4(x1)))	→	4^#(0(x1))	(169)
1^#(0(4(x1)))	→	1^#(4(x1))	(92)
4^#(0(3(5(x1))))	→	3^#(4(2(0(x1))))	(91)
0^#(5(5(3(x1))))	→	0^#(x1)	(90)
0^#(3(x1))	→	1^#(3(x1))	(87)
0^#(0(5(5(3(x1)))))	→	0^#(5(0(3(2(5(x1))))))	(163)
3^#(4(2(0(3(x1)))))	→	3^#(4(x1))	(162)
3^#(4(2(0(3(x1)))))	→	4^#(x1)	(83)
4^#(0(3(5(x1))))	→	0^#(x1)	(84)
1^#(0(4(x1)))	→	0^#(x1)	(80)
4^#(0(4(x1)))	→	4^#(0(x1))	(76)
0^#(0(4(x1)))	→	4^#(0(0(3(2(x1)))))	(73)
0^#(4(5(x1)))	→	4^#(4(0(x1)))	(74)
0^#(5(3(x1)))	→	0^#(x1)	(158)
0^#(4(3(x1)))	→	4^#(0(x1))	(72)
0^#(4(5(3(x1))))	→	4^#(x1)	(157)
0^#(0(4(x1)))	→	0^#(0(3(2(x1))))	(155)
0^#(0(5(5(3(x1)))))	→	0^#(3(2(5(x1))))	(156)
0^#(4(3(x1)))	→	4^#(x1)	(70)
0^#(1(3(x1)))	→	1^#(0(x1))	(68)
0^#(5(0(x1)))	→	4^#(0(x1))	(151)
4^#(0(4(x1)))	→	0^#(x1)	(67)
0^#(0(4(3(x1))))	→	1^#(4(x1))	(65)
0^#(4(3(x1)))	→	0^#(x1)	(150)
0^#(4(0(0(x1))))	→	1^#(0(x1))	(64)
0^#(4(3(x1)))	→	4^#(x1)	(70)
0^#(0(4(x1)))	→	0^#(3(2(x1)))	(61)
0^#(0(4(3(x1))))	→	4^#(x1)	(60)
0^#(1(3(x1)))	→	0^#(x1)	(146)
0^#(4(5(3(x1))))	→	0^#(x1)	(59)
0^#(4(3(x1)))	→	3^#(4(x1))	(58)
3^#(0(4(x1)))	→	4^#(0(3(x1)))	(141)
0^#(4(5(3(x1))))	→	3^#(5(0(2(4(x1)))))	(138)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

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

→

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

(180)

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

together with the usable rules

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

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

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

→

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

(180)

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

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

→

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

(88)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

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

→

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

(88)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

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

→

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

(123)

1.1.1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

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

→

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

(123)

could be deleted.

1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

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

→

0^#(0(0(1(4(2(x1))))))

(120)

1.1.1.1.4 Reduction Pair Processor with Usable Rules

Using the argument filter

π(0^#)	=	1
π(3^#)	=	1

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

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

and the following Max-polynomial interpretation

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

together with the usable rules

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

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

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

→

0^#(0(0(1(4(2(x1))))))

(120)

could be deleted.

1.1.1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.