Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212534)

The rewrite relation of the following TRS is considered.

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

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 150 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^#(1(4(5(0(x1)))))	→	5^#(x1)	(130)
2^#(3(3(1(5(x1)))))	→	2^#(x1)	(194)
2^#(1(4(3(5(x1)))))	→	4^#(2(x1))	(129)
0^#(1(1(2(5(x1)))))	→	5^#(1(2(x1)))	(191)
5^#(0(1(2(2(x1)))))	→	0^#(2(x1))	(123)
2^#(1(4(3(5(x1)))))	→	5^#(4(4(2(x1))))	(122)
0^#(1(2(3(5(x1)))))	→	2^#(5(x1))	(188)
5^#(1(4(2(2(x1)))))	→	2^#(4(2(x1)))	(186)
5^#(5(4(3(2(x1)))))	→	4^#(5(2(x1)))	(114)
2^#(5(3(2(x1))))	→	2^#(2(x1))	(185)
0^#(1(2(3(5(x1)))))	→	0^#(2(5(x1)))	(112)
0^#(1(4(0(2(x1)))))	→	0^#(x1)	(111)
0^#(1(2(3(5(x1)))))	→	2^#(0(x1))	(183)
0^#(1(1(5(4(x1)))))	→	5^#(x1)	(108)
0^#(1(2(3(5(x1)))))	→	5^#(2(x1))	(181)
2^#(1(1(5(x1))))	→	2^#(1(x1))	(102)
5^#(1(4(2(2(x1)))))	→	4^#(2(x1))	(176)
2^#(1(4(3(5(x1)))))	→	2^#(x1)	(94)
0^#(1(2(x1)))	→	0^#(4(x1))	(100)
2^#(2(4(3(5(x1)))))	→	2^#(x1)	(174)
2^#(3(1(1(2(x1)))))	→	2^#(0(x1))	(170)
0^#(1(1(2(5(x1)))))	→	2^#(x1)	(167)
0^#(1(2(3(5(x1)))))	→	0^#(x1)	(95)
2^#(1(4(3(5(x1)))))	→	2^#(x1)	(94)
2^#(1(4(3(5(x1)))))	→	4^#(4(2(x1)))	(92)
0^#(1(2(5(x1))))	→	2^#(x1)	(93)
0^#(1(2(x1)))	→	4^#(0(x1))	(91)
0^#(1(2(3(5(x1)))))	→	2^#(x1)	(164)
5^#(5(4(3(2(x1)))))	→	5^#(2(x1))	(161)
2^#(1(0(1(5(x1)))))	→	2^#(5(1(x1)))	(90)
0^#(1(0(1(2(x1)))))	→	0^#(x1)	(86)
2^#(3(5(5(x1))))	→	2^#(5(x1))	(157)
5^#(0(1(2(2(x1)))))	→	5^#(2(0(2(x1))))	(158)
0^#(1(2(x1)))	→	4^#(x1)	(60)
5^#(0(1(2(2(x1)))))	→	2^#(0(2(x1)))	(80)
0^#(1(1(4(2(x1)))))	→	2^#(1(x1))	(77)
2^#(3(1(1(2(x1)))))	→	2^#(1(2(0(x1))))	(152)
2^#(2(4(3(5(x1)))))	→	4^#(2(2(x1)))	(151)
0^#(1(2(3(5(x1)))))	→	4^#(5(2(x1)))	(73)
0^#(1(2(x1)))	→	0^#(x1)	(49)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
0^#(1(4(2(x1))))	→	4^#(x1)	(67)
2^#(1(0(1(5(x1)))))	→	5^#(1(x1))	(66)
2^#(5(0(3(5(x1)))))	→	0^#(5(x1))	(144)
4^#(2(0(1(2(x1)))))	→	0^#(x1)	(64)
0^#(1(4(2(x1))))	→	4^#(0(x1))	(140)
0^#(4(5(3(5(x1)))))	→	4^#(5(x1))	(138)
0^#(1(2(x1)))	→	4^#(0(4(x1)))	(139)
0^#(1(2(x1)))	→	4^#(x1)	(60)
0^#(1(2(x1)))	→	0^#(4(x1))	(100)
0^#(1(2(3(5(x1)))))	→	0^#(2(0(x1)))	(134)
2^#(2(4(3(5(x1)))))	→	2^#(2(x1))	(53)
5^#(1(4(3(2(x1)))))	→	4^#(2(x1))	(52)
2^#(3(1(1(2(x1)))))	→	0^#(x1)	(50)
0^#(1(2(x1)))	→	0^#(x1)	(49)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	x₁ + 0
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 7631
[3(x₁)]	=	x₁ + 0
[2^#(x₁)]	=	x₁ + 1
[4^#(x₁)]	=	x₁ + 0
[0(x₁)]	=	x₁ + 0
[5^#(x₁)]	=	x₁ + 7630
[2(x₁)]	=	x₁ + 1

together with the usable rules

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

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

2^#(1(4(5(0(x1)))))	→	5^#(x1)	(130)
2^#(3(3(1(5(x1)))))	→	2^#(x1)	(194)
2^#(1(4(3(5(x1)))))	→	4^#(2(x1))	(129)
0^#(1(1(2(5(x1)))))	→	5^#(1(2(x1)))	(191)
5^#(0(1(2(2(x1)))))	→	0^#(2(x1))	(123)
2^#(1(4(3(5(x1)))))	→	5^#(4(4(2(x1))))	(122)
0^#(1(2(3(5(x1)))))	→	2^#(5(x1))	(188)
5^#(1(4(2(2(x1)))))	→	2^#(4(2(x1)))	(186)
5^#(5(4(3(2(x1)))))	→	4^#(5(2(x1)))	(114)
2^#(5(3(2(x1))))	→	2^#(2(x1))	(185)
0^#(1(4(0(2(x1)))))	→	0^#(x1)	(111)
0^#(1(2(3(5(x1)))))	→	2^#(0(x1))	(183)
0^#(1(1(5(4(x1)))))	→	5^#(x1)	(108)
0^#(1(2(3(5(x1)))))	→	5^#(2(x1))	(181)
2^#(1(1(5(x1))))	→	2^#(1(x1))	(102)
5^#(1(4(2(2(x1)))))	→	4^#(2(x1))	(176)
2^#(1(4(3(5(x1)))))	→	2^#(x1)	(94)
0^#(1(2(x1)))	→	0^#(4(x1))	(100)
2^#(2(4(3(5(x1)))))	→	2^#(x1)	(174)
2^#(3(1(1(2(x1)))))	→	2^#(0(x1))	(170)
0^#(1(1(2(5(x1)))))	→	2^#(x1)	(167)
0^#(1(2(3(5(x1)))))	→	0^#(x1)	(95)
2^#(1(4(3(5(x1)))))	→	2^#(x1)	(94)
2^#(1(4(3(5(x1)))))	→	4^#(4(2(x1)))	(92)
0^#(1(2(5(x1))))	→	2^#(x1)	(93)
0^#(1(2(x1)))	→	4^#(0(x1))	(91)
0^#(1(2(3(5(x1)))))	→	2^#(x1)	(164)
5^#(5(4(3(2(x1)))))	→	5^#(2(x1))	(161)
0^#(1(0(1(2(x1)))))	→	0^#(x1)	(86)
2^#(3(5(5(x1))))	→	2^#(5(x1))	(157)
0^#(1(2(x1)))	→	4^#(x1)	(60)
5^#(0(1(2(2(x1)))))	→	2^#(0(2(x1)))	(80)
0^#(1(1(4(2(x1)))))	→	2^#(1(x1))	(77)
2^#(2(4(3(5(x1)))))	→	4^#(2(2(x1)))	(151)
0^#(1(2(3(5(x1)))))	→	4^#(5(2(x1)))	(73)
0^#(1(2(x1)))	→	0^#(x1)	(49)
0^#(1(4(2(x1))))	→	0^#(x1)	(146)
0^#(1(4(2(x1))))	→	4^#(x1)	(67)
2^#(1(0(1(5(x1)))))	→	5^#(1(x1))	(66)
2^#(5(0(3(5(x1)))))	→	0^#(5(x1))	(144)
4^#(2(0(1(2(x1)))))	→	0^#(x1)	(64)
0^#(1(4(2(x1))))	→	4^#(0(x1))	(140)
0^#(4(5(3(5(x1)))))	→	4^#(5(x1))	(138)
0^#(1(2(x1)))	→	4^#(0(4(x1)))	(139)
0^#(1(2(x1)))	→	4^#(x1)	(60)
0^#(1(2(x1)))	→	0^#(4(x1))	(100)
0^#(1(2(3(5(x1)))))	→	0^#(2(0(x1)))	(134)
2^#(2(4(3(5(x1)))))	→	2^#(2(x1))	(53)
5^#(1(4(3(2(x1)))))	→	4^#(2(x1))	(52)
2^#(3(1(1(2(x1)))))	→	0^#(x1)	(50)
0^#(1(2(x1)))	→	0^#(x1)	(49)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

0^#(1(2(3(5(x1)))))

→

0^#(2(5(x1)))

(112)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	x₁ + 1
[4(x₁)]	=	15330
[5(x₁)]	=	15331
[3(x₁)]	=	15329
[2^#(x₁)]	=	x₁ + 1
[4^#(x₁)]	=	0
[0(x₁)]	=	x₁ + 1
[5^#(x₁)]	=	x₁ + 7630
[2(x₁)]	=	15331

together with the usable rules

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

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

0^#(1(2(3(5(x1)))))

→

0^#(2(5(x1)))

(112)

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

2^#(1(0(1(5(x1)))))	→	2^#(5(1(x1)))	(90)
2^#(3(1(1(2(x1)))))	→	2^#(1(2(0(x1))))	(152)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	2
[4(x₁)]	=	1
[5(x₁)]	=	1
[3(x₁)]	=	3
[2^#(x₁)]	=	x₁ + 1
[4^#(x₁)]	=	0
[0(x₁)]	=	x₁ + 58558
[5^#(x₁)]	=	x₁ + 7630
[2(x₁)]	=	x₁ + 0

together with the usable rules

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

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

2^#(1(0(1(5(x1)))))	→	2^#(5(1(x1)))	(90)
2^#(3(1(1(2(x1)))))	→	2^#(1(2(0(x1))))	(152)

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

5^#(0(1(2(2(x1)))))

→

5^#(2(0(2(x1))))

(158)

1.1.1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 1
[1(x₁)]	=	x₁ + 1
[4(x₁)]	=	30988
[5(x₁)]	=	30989
[3(x₁)]	=	24128
[2^#(x₁)]	=	x₁ + 1
[4^#(x₁)]	=	0
[0(x₁)]	=	x₁ + 1
[5^#(x₁)]	=	x₁ + 7630
[2(x₁)]	=	30989

together with the usable rules

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

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

5^#(0(1(2(2(x1)))))

→

5^#(2(0(2(x1))))

(158)

could be deleted.

1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.