Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/214183)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

2(1(0(0(x1))))	→	0(0(1(3(2(x1)))))	(51)
2(1(0(0(x1))))	→	2(0(0(1(3(x1)))))	(52)
2(1(0(0(x1))))	→	1(1(3(2(0(0(x1))))))	(53)
4(0(1(0(x1))))	→	0(0(4(1(3(x1)))))	(54)
4(0(1(0(x1))))	→	1(3(0(0(4(3(x1))))))	(55)
4(0(1(0(x1))))	→	0(5(0(1(3(4(x1))))))	(56)
4(0(1(0(x1))))	→	0(1(3(4(0(5(x1))))))	(57)
5(0(1(0(x1))))	→	0(0(1(3(5(3(x1))))))	(58)
5(0(1(0(x1))))	→	2(0(0(1(3(5(x1))))))	(59)
2(4(1(0(x1))))	→	0(4(1(3(2(x1)))))	(60)
2(4(1(0(x1))))	→	4(2(1(3(0(4(x1))))))	(61)
2(4(1(0(x1))))	→	2(0(1(1(3(4(x1))))))	(62)
4(0(5(0(x1))))	→	5(3(0(0(4(x1)))))	(63)
4(0(5(0(x1))))	→	4(3(0(0(5(x1)))))	(64)
5(0(5(0(x1))))	→	5(0(0(2(5(3(x1))))))	(65)
5(0(5(0(x1))))	→	0(0(4(5(3(5(x1))))))	(66)
5(0(5(0(x1))))	→	0(0(4(3(5(5(x1))))))	(67)
2(1(0(2(x1))))	→	2(1(4(2(0(x1)))))	(68)
2(1(0(2(x1))))	→	1(2(2(0(5(2(x1))))))	(69)
2(1(0(2(x1))))	→	0(2(2(3(1(3(x1))))))	(70)
2(4(0(2(x1))))	→	1(2(2(0(4(x1)))))	(71)
2(4(0(2(x1))))	→	1(2(1(2(0(4(x1))))))	(72)
2(5(0(2(x1))))	→	2(2(5(1(2(0(x1))))))	(73)
2(4(0(3(x1))))	→	2(1(1(4(3(0(x1))))))	(74)
2(4(0(3(x1))))	→	2(0(1(3(3(4(x1))))))	(75)
2(2(1(0(0(x1)))))	→	1(2(1(2(0(0(x1))))))	(76)
2(3(1(0(0(x1)))))	→	4(0(2(1(3(0(x1))))))	(77)
2(3(1(0(0(x1)))))	→	0(1(3(0(5(2(x1))))))	(78)
2(2(5(0(0(x1)))))	→	1(2(5(2(0(0(x1))))))	(79)
2(5(0(1(0(x1)))))	→	5(3(0(1(2(0(x1))))))	(80)
5(5(0(1(0(x1)))))	→	5(0(0(5(3(1(x1))))))	(81)
5(0(2(1(0(x1)))))	→	0(1(2(5(0(4(x1))))))	(82)
4(0(3(1(0(x1)))))	→	0(4(4(3(0(1(x1))))))	(83)
2(4(3(1(0(x1)))))	→	4(3(0(1(2(4(x1))))))	(84)
2(2(4(1(0(x1)))))	→	2(4(2(0(1(1(x1))))))	(85)
4(2(4(1(0(x1)))))	→	4(1(3(2(0(4(x1))))))	(86)
2(3(4(1(0(x1)))))	→	4(4(0(2(1(3(x1))))))	(87)
5(0(0(3(0(x1)))))	→	0(0(0(1(3(5(x1))))))	(88)
4(0(1(4(0(x1)))))	→	0(4(1(2(0(4(x1))))))	(89)
2(2(3(4(0(x1)))))	→	2(5(3(0(2(4(x1))))))	(90)
4(0(1(5(0(x1)))))	→	0(0(1(4(2(5(x1))))))	(91)
2(4(1(5(0(x1)))))	→	2(1(5(3(0(4(x1))))))	(92)
2(1(0(1(2(x1)))))	→	3(2(0(1(1(2(x1))))))	(93)
2(5(0(4(2(x1)))))	→	2(5(3(2(4(0(x1))))))	(94)
2(1(0(0(3(x1)))))	→	2(0(1(5(3(0(x1))))))	(95)
2(5(0(0(3(x1)))))	→	1(5(3(0(0(2(x1))))))	(96)
2(4(5(0(3(x1)))))	→	1(2(5(3(0(4(x1))))))	(97)
2(1(0(2(3(x1)))))	→	2(0(1(3(1(2(x1))))))	(98)
2(5(0(2(3(x1)))))	→	5(4(3(2(0(2(x1))))))	(99)
2(5(0(2(3(x1)))))	→	2(2(2(0(3(5(x1))))))	(100)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 116 ruless (increase limit for explicit display).

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5^#(0(5(0(x1))))	→	5^#(5(x1))	(129)
5^#(0(0(3(0(x1)))))	→	5^#(x1)	(181)
5^#(0(2(1(0(x1)))))	→	2^#(5(0(4(x1))))	(166)
2^#(5(0(2(3(x1)))))	→	5^#(x1)	(213)
5^#(0(2(1(0(x1)))))	→	5^#(0(4(x1)))	(165)
5^#(0(2(1(0(x1)))))	→	4^#(x1)	(164)
4^#(0(1(5(0(x1)))))	→	4^#(2(5(x1)))	(191)
4^#(0(1(5(0(x1)))))	→	2^#(5(x1))	(190)
2^#(5(0(2(3(x1)))))	→	2^#(x1)	(209)
2^#(1(0(2(3(x1)))))	→	2^#(x1)	(207)
2^#(4(5(0(3(x1)))))	→	4^#(x1)	(204)
4^#(0(1(5(0(x1)))))	→	5^#(x1)	(189)
5^#(0(5(0(x1))))	→	5^#(x1)	(126)
5^#(0(1(0(x1))))	→	5^#(x1)	(111)
4^#(0(1(4(0(x1)))))	→	4^#(x1)	(182)
4^#(2(4(1(0(x1)))))	→	4^#(x1)	(175)
4^#(0(5(0(x1))))	→	5^#(x1)	(121)
4^#(0(5(0(x1))))	→	4^#(x1)	(119)
4^#(0(1(0(x1))))	→	4^#(0(5(x1)))	(109)
4^#(0(1(0(x1))))	→	5^#(x1)	(108)
4^#(0(1(0(x1))))	→	4^#(x1)	(106)
2^#(5(0(0(3(x1)))))	→	2^#(x1)	(202)
2^#(5(0(4(2(x1)))))	→	2^#(4(0(x1)))	(197)
2^#(5(0(4(2(x1)))))	→	4^#(0(x1))	(196)
2^#(4(1(5(0(x1)))))	→	4^#(x1)	(192)
2^#(2(3(4(0(x1)))))	→	2^#(4(x1))	(186)
2^#(2(3(4(0(x1)))))	→	4^#(x1)	(185)
2^#(4(3(1(0(x1)))))	→	2^#(4(x1))	(170)
2^#(4(3(1(0(x1)))))	→	4^#(x1)	(169)
2^#(3(1(0(0(x1)))))	→	5^#(2(x1))	(156)
2^#(3(1(0(0(x1)))))	→	2^#(x1)	(155)
2^#(4(0(3(x1))))	→	4^#(x1)	(149)
2^#(4(0(2(x1))))	→	4^#(x1)	(139)
2^#(1(0(2(x1))))	→	5^#(2(x1))	(134)
2^#(4(1(0(x1))))	→	4^#(x1)	(115)
2^#(4(1(0(x1))))	→	2^#(x1)	(113)
2^#(1(0(0(x1))))	→	2^#(x1)	(101)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the

prec(5^#)	=	stat(5^#)	=	lex
prec(4^#)	=	stat(4^#)	=	lex
prec(2^#)	=	stat(2^#)	=	lex
prec(5)	=	stat(5)	=	lex
prec(4)	=	stat(4)	=	lex
prec(3)	=	stat(3)	=	lex
prec(0)	=	stat(0)	=	lex
prec(1)	=	stat(1)	=	lex
prec(2)	=	stat(2)	=	lex

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

together with the usable rules

2(1(0(0(x1))))	→	0(0(1(3(2(x1)))))	(51)
2(1(0(0(x1))))	→	2(0(0(1(3(x1)))))	(52)
2(1(0(0(x1))))	→	1(1(3(2(0(0(x1))))))	(53)
4(0(1(0(x1))))	→	0(0(4(1(3(x1)))))	(54)
4(0(1(0(x1))))	→	1(3(0(0(4(3(x1))))))	(55)
4(0(1(0(x1))))	→	0(5(0(1(3(4(x1))))))	(56)
4(0(1(0(x1))))	→	0(1(3(4(0(5(x1))))))	(57)
5(0(1(0(x1))))	→	0(0(1(3(5(3(x1))))))	(58)
5(0(1(0(x1))))	→	2(0(0(1(3(5(x1))))))	(59)
2(4(1(0(x1))))	→	0(4(1(3(2(x1)))))	(60)
2(4(1(0(x1))))	→	4(2(1(3(0(4(x1))))))	(61)
2(4(1(0(x1))))	→	2(0(1(1(3(4(x1))))))	(62)
4(0(5(0(x1))))	→	5(3(0(0(4(x1)))))	(63)
4(0(5(0(x1))))	→	4(3(0(0(5(x1)))))	(64)
5(0(5(0(x1))))	→	5(0(0(2(5(3(x1))))))	(65)
5(0(5(0(x1))))	→	0(0(4(5(3(5(x1))))))	(66)
5(0(5(0(x1))))	→	0(0(4(3(5(5(x1))))))	(67)
2(1(0(2(x1))))	→	2(1(4(2(0(x1)))))	(68)
2(1(0(2(x1))))	→	1(2(2(0(5(2(x1))))))	(69)
2(1(0(2(x1))))	→	0(2(2(3(1(3(x1))))))	(70)
2(4(0(2(x1))))	→	1(2(2(0(4(x1)))))	(71)
2(4(0(2(x1))))	→	1(2(1(2(0(4(x1))))))	(72)
2(5(0(2(x1))))	→	2(2(5(1(2(0(x1))))))	(73)
2(4(0(3(x1))))	→	2(1(1(4(3(0(x1))))))	(74)
2(4(0(3(x1))))	→	2(0(1(3(3(4(x1))))))	(75)
2(2(1(0(0(x1)))))	→	1(2(1(2(0(0(x1))))))	(76)
2(3(1(0(0(x1)))))	→	4(0(2(1(3(0(x1))))))	(77)
2(3(1(0(0(x1)))))	→	0(1(3(0(5(2(x1))))))	(78)
2(2(5(0(0(x1)))))	→	1(2(5(2(0(0(x1))))))	(79)
2(5(0(1(0(x1)))))	→	5(3(0(1(2(0(x1))))))	(80)
5(5(0(1(0(x1)))))	→	5(0(0(5(3(1(x1))))))	(81)
5(0(2(1(0(x1)))))	→	0(1(2(5(0(4(x1))))))	(82)
4(0(3(1(0(x1)))))	→	0(4(4(3(0(1(x1))))))	(83)
2(4(3(1(0(x1)))))	→	4(3(0(1(2(4(x1))))))	(84)
2(2(4(1(0(x1)))))	→	2(4(2(0(1(1(x1))))))	(85)
4(2(4(1(0(x1)))))	→	4(1(3(2(0(4(x1))))))	(86)
2(3(4(1(0(x1)))))	→	4(4(0(2(1(3(x1))))))	(87)
5(0(0(3(0(x1)))))	→	0(0(0(1(3(5(x1))))))	(88)
4(0(1(4(0(x1)))))	→	0(4(1(2(0(4(x1))))))	(89)
2(2(3(4(0(x1)))))	→	2(5(3(0(2(4(x1))))))	(90)
4(0(1(5(0(x1)))))	→	0(0(1(4(2(5(x1))))))	(91)
2(4(1(5(0(x1)))))	→	2(1(5(3(0(4(x1))))))	(92)
2(1(0(1(2(x1)))))	→	3(2(0(1(1(2(x1))))))	(93)
2(5(0(4(2(x1)))))	→	2(5(3(2(4(0(x1))))))	(94)
2(1(0(0(3(x1)))))	→	2(0(1(5(3(0(x1))))))	(95)
2(5(0(0(3(x1)))))	→	1(5(3(0(0(2(x1))))))	(96)
2(4(5(0(3(x1)))))	→	1(2(5(3(0(4(x1))))))	(97)
2(1(0(2(3(x1)))))	→	2(0(1(3(1(2(x1))))))	(98)
2(5(0(2(3(x1)))))	→	5(4(3(2(0(2(x1))))))	(99)
2(5(0(2(3(x1)))))	→	2(2(2(0(3(5(x1))))))	(100)

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

5^#(0(5(0(x1))))	→	5^#(5(x1))	(129)
5^#(0(0(3(0(x1)))))	→	5^#(x1)	(181)
5^#(0(2(1(0(x1)))))	→	2^#(5(0(4(x1))))	(166)
2^#(5(0(2(3(x1)))))	→	5^#(x1)	(213)
5^#(0(2(1(0(x1)))))	→	5^#(0(4(x1)))	(165)
5^#(0(2(1(0(x1)))))	→	4^#(x1)	(164)
4^#(0(1(5(0(x1)))))	→	4^#(2(5(x1)))	(191)
4^#(0(1(5(0(x1)))))	→	2^#(5(x1))	(190)
2^#(5(0(2(3(x1)))))	→	2^#(x1)	(209)
2^#(1(0(2(3(x1)))))	→	2^#(x1)	(207)
2^#(4(5(0(3(x1)))))	→	4^#(x1)	(204)
4^#(0(1(5(0(x1)))))	→	5^#(x1)	(189)
5^#(0(5(0(x1))))	→	5^#(x1)	(126)
5^#(0(1(0(x1))))	→	5^#(x1)	(111)
4^#(0(1(4(0(x1)))))	→	4^#(x1)	(182)
4^#(2(4(1(0(x1)))))	→	4^#(x1)	(175)
4^#(0(5(0(x1))))	→	5^#(x1)	(121)
4^#(0(5(0(x1))))	→	4^#(x1)	(119)
4^#(0(1(0(x1))))	→	4^#(0(5(x1)))	(109)
4^#(0(1(0(x1))))	→	5^#(x1)	(108)
4^#(0(1(0(x1))))	→	4^#(x1)	(106)
2^#(5(0(0(3(x1)))))	→	2^#(x1)	(202)
2^#(4(1(5(0(x1)))))	→	4^#(x1)	(192)
2^#(2(3(4(0(x1)))))	→	2^#(4(x1))	(186)
2^#(2(3(4(0(x1)))))	→	4^#(x1)	(185)
2^#(4(3(1(0(x1)))))	→	2^#(4(x1))	(170)
2^#(4(3(1(0(x1)))))	→	4^#(x1)	(169)
2^#(3(1(0(0(x1)))))	→	5^#(2(x1))	(156)
2^#(3(1(0(0(x1)))))	→	2^#(x1)	(155)
2^#(4(0(3(x1))))	→	4^#(x1)	(149)
2^#(4(0(2(x1))))	→	4^#(x1)	(139)
2^#(1(0(2(x1))))	→	5^#(2(x1))	(134)
2^#(4(1(0(x1))))	→	4^#(x1)	(115)
2^#(4(1(0(x1))))	→	2^#(x1)	(113)
2^#(1(0(0(x1))))	→	2^#(x1)	(101)

could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

2^#(5(0(4(2(x1)))))

→

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

(197)

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

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

together with the usable rules

2(1(0(0(x1))))	→	0(0(1(3(2(x1)))))	(51)
2(1(0(0(x1))))	→	2(0(0(1(3(x1)))))	(52)
2(1(0(0(x1))))	→	1(1(3(2(0(0(x1))))))	(53)
4(0(1(0(x1))))	→	0(0(4(1(3(x1)))))	(54)
4(0(1(0(x1))))	→	1(3(0(0(4(3(x1))))))	(55)
4(0(1(0(x1))))	→	0(5(0(1(3(4(x1))))))	(56)
4(0(1(0(x1))))	→	0(1(3(4(0(5(x1))))))	(57)
5(0(1(0(x1))))	→	0(0(1(3(5(3(x1))))))	(58)
5(0(1(0(x1))))	→	2(0(0(1(3(5(x1))))))	(59)
2(4(1(0(x1))))	→	0(4(1(3(2(x1)))))	(60)
2(4(1(0(x1))))	→	4(2(1(3(0(4(x1))))))	(61)
2(4(1(0(x1))))	→	2(0(1(1(3(4(x1))))))	(62)
4(0(5(0(x1))))	→	5(3(0(0(4(x1)))))	(63)
4(0(5(0(x1))))	→	4(3(0(0(5(x1)))))	(64)
5(0(5(0(x1))))	→	5(0(0(2(5(3(x1))))))	(65)
5(0(5(0(x1))))	→	0(0(4(5(3(5(x1))))))	(66)
5(0(5(0(x1))))	→	0(0(4(3(5(5(x1))))))	(67)
2(1(0(2(x1))))	→	2(1(4(2(0(x1)))))	(68)
2(1(0(2(x1))))	→	1(2(2(0(5(2(x1))))))	(69)
2(1(0(2(x1))))	→	0(2(2(3(1(3(x1))))))	(70)
2(4(0(2(x1))))	→	1(2(2(0(4(x1)))))	(71)
2(4(0(2(x1))))	→	1(2(1(2(0(4(x1))))))	(72)
2(5(0(2(x1))))	→	2(2(5(1(2(0(x1))))))	(73)
2(4(0(3(x1))))	→	2(1(1(4(3(0(x1))))))	(74)
2(4(0(3(x1))))	→	2(0(1(3(3(4(x1))))))	(75)
2(2(1(0(0(x1)))))	→	1(2(1(2(0(0(x1))))))	(76)
2(3(1(0(0(x1)))))	→	4(0(2(1(3(0(x1))))))	(77)
2(3(1(0(0(x1)))))	→	0(1(3(0(5(2(x1))))))	(78)
2(2(5(0(0(x1)))))	→	1(2(5(2(0(0(x1))))))	(79)
2(5(0(1(0(x1)))))	→	5(3(0(1(2(0(x1))))))	(80)
5(5(0(1(0(x1)))))	→	5(0(0(5(3(1(x1))))))	(81)
5(0(2(1(0(x1)))))	→	0(1(2(5(0(4(x1))))))	(82)
4(0(3(1(0(x1)))))	→	0(4(4(3(0(1(x1))))))	(83)
2(4(3(1(0(x1)))))	→	4(3(0(1(2(4(x1))))))	(84)
2(2(4(1(0(x1)))))	→	2(4(2(0(1(1(x1))))))	(85)
4(2(4(1(0(x1)))))	→	4(1(3(2(0(4(x1))))))	(86)
2(3(4(1(0(x1)))))	→	4(4(0(2(1(3(x1))))))	(87)
5(0(0(3(0(x1)))))	→	0(0(0(1(3(5(x1))))))	(88)
4(0(1(4(0(x1)))))	→	0(4(1(2(0(4(x1))))))	(89)
2(2(3(4(0(x1)))))	→	2(5(3(0(2(4(x1))))))	(90)
4(0(1(5(0(x1)))))	→	0(0(1(4(2(5(x1))))))	(91)
2(4(1(5(0(x1)))))	→	2(1(5(3(0(4(x1))))))	(92)
2(1(0(1(2(x1)))))	→	3(2(0(1(1(2(x1))))))	(93)
2(5(0(4(2(x1)))))	→	2(5(3(2(4(0(x1))))))	(94)
2(1(0(0(3(x1)))))	→	2(0(1(5(3(0(x1))))))	(95)
2(5(0(0(3(x1)))))	→	1(5(3(0(0(2(x1))))))	(96)
2(4(5(0(3(x1)))))	→	1(2(5(3(0(4(x1))))))	(97)
2(1(0(2(3(x1)))))	→	2(0(1(3(1(2(x1))))))	(98)
2(5(0(2(3(x1)))))	→	5(4(3(2(0(2(x1))))))	(99)
2(5(0(2(3(x1)))))	→	2(2(2(0(3(5(x1))))))	(100)

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

2^#(5(0(4(2(x1)))))

→

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

(197)

could be deleted.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.