Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213537)

The rewrite relation of the following TRS is considered.

0(1(0(0(x1))))	→	2(0(2(0(0(1(x1))))))	(1)
1(0(0(1(x1))))	→	1(0(3(0(1(x1)))))	(2)
1(0(0(4(x1))))	→	1(4(0(2(0(x1)))))	(3)
1(0(0(4(x1))))	→	1(4(0(3(2(0(x1))))))	(4)
1(0(1(0(x1))))	→	1(3(0(0(3(1(x1))))))	(5)
1(0(1(4(x1))))	→	1(0(3(1(2(4(x1))))))	(6)
1(0(1(5(x1))))	→	1(0(2(1(5(x1)))))	(7)
1(0(4(1(x1))))	→	1(0(3(4(1(x1)))))	(8)
1(0(5(0(x1))))	→	1(2(5(2(0(0(x1))))))	(9)
1(2(0(4(x1))))	→	1(3(4(2(0(x1)))))	(10)
1(3(0(4(x1))))	→	1(3(4(2(0(x1)))))	(11)
1(4(1(0(x1))))	→	1(4(0(2(1(x1)))))	(12)
1(4(1(0(x1))))	→	1(4(0(3(1(x1)))))	(13)
1(4(5(0(x1))))	→	4(0(2(1(2(5(x1))))))	(14)
1(5(0(1(x1))))	→	0(1(3(1(3(5(x1))))))	(15)
1(5(0(1(x1))))	→	1(5(0(3(1(3(x1))))))	(16)
4(1(0(0(x1))))	→	0(2(4(2(0(1(x1))))))	(17)
4(1(0(0(x1))))	→	0(3(4(3(0(1(x1))))))	(18)
4(1(0(5(x1))))	→	4(0(2(1(5(x1)))))	(19)
4(1(2(0(x1))))	→	3(0(2(1(3(4(x1))))))	(20)
4(5(0(0(x1))))	→	3(0(4(0(5(x1)))))	(21)
5(0(0(4(x1))))	→	5(4(0(2(0(x1)))))	(22)
5(0(1(4(x1))))	→	1(5(4(3(0(x1)))))	(23)
5(4(1(5(x1))))	→	4(3(1(5(5(x1)))))	(24)
1(0(1(4(5(x1)))))	→	1(3(1(5(4(0(x1))))))	(25)
1(0(4(0(4(x1)))))	→	1(0(4(3(4(0(x1))))))	(26)
1(1(1(0(0(x1)))))	→	1(1(0(0(3(1(x1))))))	(27)
1(1(4(5(0(x1)))))	→	1(1(3(4(0(5(x1))))))	(28)
1(2(0(5(0(x1)))))	→	5(2(1(0(2(0(x1))))))	(29)
1(4(5(0(0(x1)))))	→	4(0(5(0(2(1(x1))))))	(30)
1(4(5(2(0(x1)))))	→	0(3(1(2(5(4(x1))))))	(31)
1(5(0(4(5(x1)))))	→	3(4(0(5(1(5(x1))))))	(32)
4(1(0(0(1(x1)))))	→	1(0(4(0(3(1(x1))))))	(33)
4(1(2(0(0(x1)))))	→	4(2(0(0(2(1(x1))))))	(34)
4(1(3(0(5(x1)))))	→	3(2(4(0(1(5(x1))))))	(35)
4(1(5(3(1(x1)))))	→	2(1(5(4(3(1(x1))))))	(36)
4(4(1(0(1(x1)))))	→	1(3(4(4(0(1(x1))))))	(37)
4(5(0(1(0(x1)))))	→	4(0(0(5(3(1(x1))))))	(38)
4(5(1(1(0(x1)))))	→	1(1(3(4(0(5(x1))))))	(39)
4(5(1(1(0(x1)))))	→	4(0(2(1(5(1(x1))))))	(40)
4(5(1(5(1(x1)))))	→	4(1(5(5(3(1(x1))))))	(41)
4(5(5(0(4(x1)))))	→	5(4(0(5(4(2(x1))))))	(42)
5(0(0(4(1(x1)))))	→	5(4(0(3(0(1(x1))))))	(43)
5(0(0(4(5(x1)))))	→	0(3(4(0(5(5(x1))))))	(44)
5(0(0(4(5(x1)))))	→	4(0(0(3(5(5(x1))))))	(45)
5(0(4(1(0(x1)))))	→	5(0(4(0(3(1(x1))))))	(46)
5(1(5(0(1(x1)))))	→	5(0(3(5(1(1(x1))))))	(47)
5(4(1(1(0(x1)))))	→	1(2(4(0(1(5(x1))))))	(48)
5(4(5(1(0(x1)))))	→	5(5(1(3(4(0(x1))))))	(49)
5(5(2(0(4(x1)))))	→	5(5(3(4(2(0(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 194 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

1^#(5(0(1(x1))))	→	5^#(x1)	(239)
4^#(5(1(1(0(x1)))))	→	5^#(x1)	(150)
1^#(4(5(2(0(x1)))))	→	4^#(x1)	(236)
4^#(5(1(1(0(x1)))))	→	5^#(1(x1))	(237)
4^#(5(1(1(0(x1)))))	→	1^#(1(3(4(0(5(x1))))))	(235)
4^#(5(1(1(0(x1)))))	→	1^#(5(1(x1)))	(146)
4^#(5(1(1(0(x1)))))	→	4^#(0(5(x1)))	(141)
1^#(2(0(4(x1))))	→	0^#(x1)	(139)
4^#(1(0(0(x1))))	→	1^#(x1)	(184)
1^#(4(5(0(0(x1)))))	→	1^#(x1)	(136)
1^#(5(0(1(x1))))	→	1^#(3(1(3(5(x1)))))	(228)
1^#(4(5(2(0(x1)))))	→	5^#(4(x1))	(134)
4^#(5(0(0(x1))))	→	5^#(x1)	(133)
5^#(0(0(4(5(x1)))))	→	4^#(0(5(5(x1))))	(227)
4^#(1(3(0(5(x1)))))	→	1^#(5(x1))	(131)
1^#(1(1(0(0(x1)))))	→	1^#(x1)	(226)
4^#(1(2(0(x1))))	→	4^#(x1)	(130)
1^#(5(0(1(x1))))	→	1^#(3(x1))	(225)
4^#(1(0(5(x1))))	→	1^#(5(x1))	(224)
1^#(0(1(4(5(x1)))))	→	5^#(4(0(x1)))	(128)
1^#(4(5(2(0(x1)))))	→	1^#(2(5(4(x1))))	(126)
1^#(1(4(5(0(x1)))))	→	1^#(1(3(4(0(5(x1))))))	(220)
1^#(1(4(5(0(x1)))))	→	5^#(x1)	(123)
5^#(5(2(0(4(x1)))))	→	0^#(x1)	(116)
1^#(0(1(4(5(x1)))))	→	1^#(3(1(5(4(0(x1))))))	(118)
1^#(4(1(0(x1))))	→	1^#(x1)	(102)
5^#(4(1(1(0(x1)))))	→	0^#(1(5(x1)))	(212)
1^#(0(4(0(4(x1)))))	→	4^#(0(x1))	(113)
5^#(0(1(4(x1))))	→	0^#(x1)	(210)
1^#(5(0(1(x1))))	→	1^#(3(5(x1)))	(208)
5^#(0(0(4(5(x1)))))	→	0^#(5(5(x1)))	(209)
4^#(1(2(0(x1))))	→	1^#(3(4(x1)))	(206)
1^#(0(1(4(5(x1)))))	→	0^#(x1)	(112)
4^#(5(1(1(0(x1)))))	→	1^#(3(4(0(5(x1)))))	(204)
4^#(1(3(0(5(x1)))))	→	0^#(1(5(x1)))	(111)
5^#(4(1(5(x1))))	→	1^#(5(5(x1)))	(202)
1^#(0(0(4(x1))))	→	0^#(x1)	(194)
0^#(1(0(0(x1))))	→	1^#(x1)	(201)
4^#(5(1(1(0(x1)))))	→	0^#(5(x1))	(107)
4^#(5(1(1(0(x1)))))	→	1^#(x1)	(108)
1^#(5(0(4(5(x1)))))	→	4^#(0(5(1(5(x1)))))	(106)
1^#(1(4(5(0(x1)))))	→	1^#(3(4(0(5(x1)))))	(199)
5^#(1(5(0(1(x1)))))	→	5^#(1(1(x1)))	(196)
5^#(4(5(1(0(x1)))))	→	5^#(5(1(3(4(0(x1))))))	(105)
1^#(0(0(4(x1))))	→	0^#(x1)	(194)
5^#(0(4(1(0(x1)))))	→	1^#(x1)	(195)
5^#(0(0(4(x1))))	→	0^#(x1)	(101)
1^#(4(1(0(x1))))	→	1^#(x1)	(102)
4^#(1(0(0(x1))))	→	0^#(1(x1))	(99)
4^#(5(0(0(x1))))	→	0^#(4(0(5(x1))))	(189)
5^#(0(0(4(1(x1)))))	→	0^#(1(x1))	(96)
4^#(1(0(0(x1))))	→	0^#(1(x1))	(99)
4^#(4(1(0(1(x1)))))	→	1^#(3(4(4(0(1(x1))))))	(186)
4^#(1(0(0(x1))))	→	1^#(x1)	(184)
1^#(0(4(0(4(x1)))))	→	0^#(x1)	(185)
1^#(3(0(4(x1))))	→	0^#(x1)	(90)
5^#(4(1(1(0(x1)))))	→	5^#(x1)	(91)
1^#(0(5(0(x1))))	→	0^#(0(x1))	(86)
1^#(4(5(0(x1))))	→	5^#(x1)	(182)
5^#(4(1(1(0(x1)))))	→	1^#(2(4(0(1(5(x1))))))	(179)
5^#(0(0(4(5(x1)))))	→	5^#(5(x1))	(80)
1^#(5(0(4(5(x1)))))	→	0^#(5(1(5(x1))))	(77)
1^#(1(4(5(0(x1)))))	→	0^#(5(x1))	(177)
1^#(5(0(4(5(x1)))))	→	5^#(1(5(x1)))	(174)
4^#(4(1(0(1(x1)))))	→	4^#(4(0(1(x1))))	(175)
5^#(4(5(1(0(x1)))))	→	5^#(1(3(4(0(x1)))))	(70)
1^#(0(1(4(5(x1)))))	→	4^#(0(x1))	(69)
4^#(5(0(0(x1))))	→	0^#(5(x1))	(68)
4^#(5(0(0(x1))))	→	4^#(0(5(x1)))	(168)
5^#(0(0(4(5(x1)))))	→	5^#(5(x1))	(80)
5^#(4(1(1(0(x1)))))	→	1^#(5(x1))	(65)
5^#(4(1(5(x1))))	→	5^#(5(x1))	(166)
5^#(1(5(0(1(x1)))))	→	1^#(1(x1))	(167)
0^#(1(0(0(x1))))	→	0^#(1(x1))	(163)
1^#(0(1(0(x1))))	→	1^#(x1)	(164)
5^#(4(5(1(0(x1)))))	→	4^#(0(x1))	(165)
1^#(5(0(4(5(x1)))))	→	1^#(5(x1))	(62)
1^#(0(1(4(5(x1)))))	→	1^#(5(4(0(x1))))	(63)
5^#(4(5(1(0(x1)))))	→	1^#(3(4(0(x1))))	(162)
4^#(1(3(0(5(x1)))))	→	4^#(0(1(5(x1))))	(161)
4^#(1(2(0(0(x1)))))	→	1^#(x1)	(160)
1^#(0(1(4(x1))))	→	1^#(2(4(x1)))	(59)
1^#(4(5(0(x1))))	→	1^#(2(5(x1)))	(56)
5^#(4(1(1(0(x1)))))	→	4^#(0(1(5(x1))))	(55)
0^#(1(0(0(x1))))	→	0^#(0(1(x1)))	(155)
4^#(4(1(0(1(x1)))))	→	4^#(0(1(x1)))	(54)
1^#(5(0(1(x1))))	→	0^#(1(3(1(3(5(x1))))))	(153)
1^#(1(4(5(0(x1)))))	→	4^#(0(5(x1)))	(154)
4^#(5(0(1(0(x1)))))	→	1^#(x1)	(53)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	x₁ + 1
[4(x₁)]	=	x₁ + 41684
[5(x₁)]	=	x₁ + 1
[3(x₁)]	=	x₁ + 0
[4^#(x₁)]	=	x₁ + 41685
[0(x₁)]	=	x₁ + 1
[5^#(x₁)]	=	x₁ + 3
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	x₁ + 2

together with the usable rules

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

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

1^#(5(0(1(x1))))	→	5^#(x1)	(239)
4^#(5(1(1(0(x1)))))	→	5^#(x1)	(150)
1^#(4(5(2(0(x1)))))	→	4^#(x1)	(236)
4^#(5(1(1(0(x1)))))	→	5^#(1(x1))	(237)
4^#(5(1(1(0(x1)))))	→	1^#(5(1(x1)))	(146)
4^#(5(1(1(0(x1)))))	→	4^#(0(5(x1)))	(141)
1^#(2(0(4(x1))))	→	0^#(x1)	(139)
4^#(1(0(0(x1))))	→	1^#(x1)	(184)
1^#(4(5(0(0(x1)))))	→	1^#(x1)	(136)
1^#(5(0(1(x1))))	→	1^#(3(1(3(5(x1)))))	(228)
1^#(4(5(2(0(x1)))))	→	5^#(4(x1))	(134)
4^#(5(0(0(x1))))	→	5^#(x1)	(133)
5^#(0(0(4(5(x1)))))	→	4^#(0(5(5(x1))))	(227)
4^#(1(3(0(5(x1)))))	→	1^#(5(x1))	(131)
1^#(1(1(0(0(x1)))))	→	1^#(x1)	(226)
4^#(1(2(0(x1))))	→	4^#(x1)	(130)
1^#(5(0(1(x1))))	→	1^#(3(x1))	(225)
4^#(1(0(5(x1))))	→	1^#(5(x1))	(224)
1^#(0(1(4(5(x1)))))	→	5^#(4(0(x1)))	(128)
1^#(4(5(2(0(x1)))))	→	1^#(2(5(4(x1))))	(126)
1^#(1(4(5(0(x1)))))	→	5^#(x1)	(123)
5^#(5(2(0(4(x1)))))	→	0^#(x1)	(116)
1^#(4(1(0(x1))))	→	1^#(x1)	(102)
5^#(4(1(1(0(x1)))))	→	0^#(1(5(x1)))	(212)
1^#(0(4(0(4(x1)))))	→	4^#(0(x1))	(113)
5^#(0(1(4(x1))))	→	0^#(x1)	(210)
1^#(5(0(1(x1))))	→	1^#(3(5(x1)))	(208)
5^#(0(0(4(5(x1)))))	→	0^#(5(5(x1)))	(209)
4^#(1(2(0(x1))))	→	1^#(3(4(x1)))	(206)
1^#(0(1(4(5(x1)))))	→	0^#(x1)	(112)
4^#(5(1(1(0(x1)))))	→	1^#(3(4(0(5(x1)))))	(204)
4^#(1(3(0(5(x1)))))	→	0^#(1(5(x1)))	(111)
5^#(4(1(5(x1))))	→	1^#(5(5(x1)))	(202)
1^#(0(0(4(x1))))	→	0^#(x1)	(194)
0^#(1(0(0(x1))))	→	1^#(x1)	(201)
4^#(5(1(1(0(x1)))))	→	0^#(5(x1))	(107)
4^#(5(1(1(0(x1)))))	→	1^#(x1)	(108)
1^#(1(4(5(0(x1)))))	→	1^#(3(4(0(5(x1)))))	(199)
5^#(1(5(0(1(x1)))))	→	5^#(1(1(x1)))	(196)
1^#(0(0(4(x1))))	→	0^#(x1)	(194)
5^#(0(4(1(0(x1)))))	→	1^#(x1)	(195)
5^#(0(0(4(x1))))	→	0^#(x1)	(101)
1^#(4(1(0(x1))))	→	1^#(x1)	(102)
4^#(1(0(0(x1))))	→	0^#(1(x1))	(99)
4^#(5(0(0(x1))))	→	0^#(4(0(5(x1))))	(189)
5^#(0(0(4(1(x1)))))	→	0^#(1(x1))	(96)
4^#(1(0(0(x1))))	→	0^#(1(x1))	(99)
4^#(1(0(0(x1))))	→	1^#(x1)	(184)
1^#(0(4(0(4(x1)))))	→	0^#(x1)	(185)
1^#(3(0(4(x1))))	→	0^#(x1)	(90)
5^#(4(1(1(0(x1)))))	→	5^#(x1)	(91)
1^#(0(5(0(x1))))	→	0^#(0(x1))	(86)
1^#(4(5(0(x1))))	→	5^#(x1)	(182)
5^#(4(1(1(0(x1)))))	→	1^#(2(4(0(1(5(x1))))))	(179)
5^#(0(0(4(5(x1)))))	→	5^#(5(x1))	(80)
1^#(5(0(4(5(x1)))))	→	0^#(5(1(5(x1))))	(77)
1^#(1(4(5(0(x1)))))	→	0^#(5(x1))	(177)
1^#(5(0(4(5(x1)))))	→	5^#(1(5(x1)))	(174)
4^#(4(1(0(1(x1)))))	→	4^#(4(0(1(x1))))	(175)
5^#(4(5(1(0(x1)))))	→	5^#(1(3(4(0(x1)))))	(70)
1^#(0(1(4(5(x1)))))	→	4^#(0(x1))	(69)
4^#(5(0(0(x1))))	→	0^#(5(x1))	(68)
4^#(5(0(0(x1))))	→	4^#(0(5(x1)))	(168)
5^#(0(0(4(5(x1)))))	→	5^#(5(x1))	(80)
5^#(4(1(1(0(x1)))))	→	1^#(5(x1))	(65)
5^#(4(1(5(x1))))	→	5^#(5(x1))	(166)
5^#(1(5(0(1(x1)))))	→	1^#(1(x1))	(167)
0^#(1(0(0(x1))))	→	0^#(1(x1))	(163)
1^#(0(1(0(x1))))	→	1^#(x1)	(164)
5^#(4(5(1(0(x1)))))	→	4^#(0(x1))	(165)
1^#(5(0(4(5(x1)))))	→	1^#(5(x1))	(62)
1^#(0(1(4(5(x1)))))	→	1^#(5(4(0(x1))))	(63)
5^#(4(5(1(0(x1)))))	→	1^#(3(4(0(x1))))	(162)
4^#(1(2(0(0(x1)))))	→	1^#(x1)	(160)
1^#(0(1(4(x1))))	→	1^#(2(4(x1)))	(59)
1^#(4(5(0(x1))))	→	1^#(2(5(x1)))	(56)
5^#(4(1(1(0(x1)))))	→	4^#(0(1(5(x1))))	(55)
0^#(1(0(0(x1))))	→	0^#(0(1(x1)))	(155)
4^#(4(1(0(1(x1)))))	→	4^#(0(1(x1)))	(54)
1^#(5(0(1(x1))))	→	0^#(1(3(1(3(5(x1))))))	(153)
1^#(1(4(5(0(x1)))))	→	4^#(0(5(x1)))	(154)
4^#(5(0(1(0(x1)))))	→	1^#(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

5^#(4(5(1(0(x1)))))

→

5^#(5(1(3(4(0(x1))))))

(105)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	x₁ + 1
[4(x₁)]	=	x₁ + 1
[5(x₁)]	=	x₁ + 1
[3(x₁)]	=	1
[4^#(x₁)]	=	x₁ + 41685
[0(x₁)]	=	x₁ + 1
[5^#(x₁)]	=	x₁ + 3
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	x₁ + 2

together with the usable rules

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

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

5^#(4(5(1(0(x1)))))

→

5^#(5(1(3(4(0(x1))))))

(105)

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

1^#(5(0(4(5(x1)))))	→	4^#(0(5(1(5(x1)))))	(106)
1^#(1(4(5(0(x1)))))	→	1^#(1(3(4(0(5(x1))))))	(220)
4^#(5(1(1(0(x1)))))	→	1^#(1(3(4(0(5(x1))))))	(235)
4^#(1(3(0(5(x1)))))	→	4^#(0(1(5(x1))))	(161)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	x₁ + 1
[4(x₁)]	=	x₁ + 0
[5(x₁)]	=	x₁ + 0
[3(x₁)]	=	1
[4^#(x₁)]	=	x₁ + 41685
[0(x₁)]	=	2
[5^#(x₁)]	=	x₁ + 3
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	x₁ + 41686

together with the usable rules

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

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

1^#(5(0(4(5(x1)))))	→	4^#(0(5(1(5(x1)))))	(106)
1^#(1(4(5(0(x1)))))	→	1^#(1(3(4(0(5(x1))))))	(220)
4^#(5(1(1(0(x1)))))	→	1^#(1(3(4(0(5(x1))))))	(235)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

4^#(1(3(0(5(x1)))))

→

4^#(0(1(5(x1))))

(161)

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	2
[4(x₁)]	=	x₁ + 1
[5(x₁)]	=	2
[3(x₁)]	=	1
[4^#(x₁)]	=	x₁ + 41685
[0(x₁)]	=	1
[5^#(x₁)]	=	x₁ + 3
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	x₁ + 41686

together with the usable rules

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

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

4^#(1(3(0(5(x1)))))

→

4^#(0(1(5(x1))))

(161)

could be deleted.

1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.