Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/85650)

The rewrite relation of the following TRS is considered.

0(0(1(0(x1))))	→	2(3(4(x1)))	(1)
3(5(1(1(2(1(x1))))))	→	3(3(3(1(3(1(x1))))))	(2)
0(0(3(3(5(5(4(x1)))))))	→	2(3(3(0(3(4(x1))))))	(3)
0(4(0(0(4(4(5(x1)))))))	→	3(5(4(1(2(4(x1))))))	(4)
5(5(5(2(5(2(4(4(3(x1)))))))))	→	5(0(1(5(0(4(5(2(x1))))))))	(5)
0(4(3(0(1(1(1(4(1(0(x1))))))))))	→	2(5(0(2(2(1(0(5(1(3(x1))))))))))	(6)
0(4(5(2(1(5(3(0(1(1(x1))))))))))	→	5(3(0(2(3(0(5(0(0(0(x1))))))))))	(7)
3(3(1(4(2(0(3(5(0(0(x1))))))))))	→	0(1(1(2(5(3(1(2(2(2(x1))))))))))	(8)
4(5(2(2(1(5(2(4(5(0(x1))))))))))	→	4(2(5(4(1(4(5(5(5(0(x1))))))))))	(9)
0(1(1(3(4(4(0(4(1(5(5(x1)))))))))))	→	0(2(4(4(5(5(5(3(2(4(0(x1)))))))))))	(10)
2(1(1(3(5(5(4(1(0(4(4(1(x1))))))))))))	→	2(5(5(3(4(2(0(4(0(1(2(0(x1))))))))))))	(11)
4(4(5(1(1(1(3(2(5(5(4(1(x1))))))))))))	→	4(2(1(0(2(0(0(0(4(3(3(1(x1))))))))))))	(12)
4(5(2(1(0(5(2(0(2(5(0(4(x1))))))))))))	→	3(0(0(5(3(2(0(0(2(0(0(4(x1))))))))))))	(13)
5(2(2(4(2(5(1(4(5(4(0(4(x1))))))))))))	→	2(4(5(5(5(4(3(2(5(3(1(4(x1))))))))))))	(14)
0(4(1(5(4(3(5(5(0(0(5(1(2(x1)))))))))))))	→	0(3(5(4(5(0(5(2(0(4(2(4(x1))))))))))))	(15)
2(2(2(1(0(2(2(5(1(0(1(5(1(x1)))))))))))))	→	5(4(0(3(2(5(0(1(2(4(3(3(x1))))))))))))	(16)
0(2(5(5(1(2(2(5(5(0(5(2(1(0(x1))))))))))))))	→	2(0(5(5(5(0(1(2(3(4(0(1(1(x1)))))))))))))	(17)
2(4(1(3(4(1(0(0(3(2(4(4(1(2(x1))))))))))))))	→	3(4(5(1(4(4(4(5(4(5(1(4(3(1(x1))))))))))))))	(18)
0(3(3(2(0(2(2(3(4(0(5(1(2(5(1(2(x1))))))))))))))))	→	0(1(0(1(3(0(1(1(2(2(4(0(3(4(4(x1)))))))))))))))	(19)
2(2(0(0(2(0(5(2(4(0(4(3(2(5(2(0(x1))))))))))))))))	→	4(5(4(1(3(5(1(3(0(2(5(0(2(5(3(x1)))))))))))))))	(20)
1(4(4(3(4(5(3(5(1(2(5(0(4(4(4(0(4(x1)))))))))))))))))	→	1(2(3(3(1(0(1(1(1(0(3(3(3(3(5(3(4(x1)))))))))))))))))	(21)
3(0(2(0(3(2(3(3(0(4(3(0(0(2(0(2(0(x1)))))))))))))))))	→	3(5(5(3(2(1(2(2(2(0(3(0(4(5(1(1(x1))))))))))))))))	(22)
4(0(4(3(3(5(2(1(3(5(5(3(2(4(5(1(0(x1)))))))))))))))))	→	4(4(5(5(4(2(2(0(2(1(5(2(1(5(4(5(0(x1)))))))))))))))))	(23)
1(2(4(3(4(4(2(2(1(4(2(1(3(1(5(1(5(3(x1))))))))))))))))))	→	1(0(3(2(0(0(1(3(0(3(4(0(0(0(3(5(3(x1)))))))))))))))))	(24)
4(5(5(5(4(0(0(3(3(4(4(3(3(5(5(3(1(0(x1))))))))))))))))))	→	3(3(2(5(1(1(5(4(1(5(1(2(5(4(1(2(1(x1)))))))))))))))))	(25)
1(1(1(3(0(3(5(0(5(1(1(4(2(2(0(3(2(3(3(x1)))))))))))))))))))	→	3(5(2(2(4(4(1(2(4(3(0(5(5(0(0(1(2(2(3(x1)))))))))))))))))))	(26)
0(2(1(3(1(2(0(3(4(2(2(5(3(0(1(3(3(5(3(2(x1))))))))))))))))))))	→	1(3(1(0(2(5(3(1(1(3(1(5(0(1(1(0(3(0(3(x1)))))))))))))))))))	(27)
1(0(0(3(2(0(4(5(1(1(2(5(0(1(0(2(1(4(3(2(x1))))))))))))))))))))	→	3(1(1(4(4(5(4(3(1(0(3(0(4(0(1(0(2(2(2(1(x1))))))))))))))))))))	(28)
0(0(3(3(3(1(1(0(0(2(5(0(4(5(1(4(2(4(1(0(5(x1)))))))))))))))))))))	→	1(2(3(0(5(2(3(0(3(5(4(1(4(4(5(0(1(1(1(5(x1))))))))))))))))))))	(29)
0(5(3(1(0(2(1(4(5(1(1(0(1(1(3(2(5(4(3(0(4(x1)))))))))))))))))))))	→	1(1(5(5(4(2(3(5(3(1(0(0(5(3(5(5(4(1(3(4(x1))))))))))))))))))))	(30)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

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

0(1(0(0(x1))))	→	4(3(2(x1)))	(31)
1(2(1(1(5(3(x1))))))	→	1(3(1(3(3(3(x1))))))	(32)
4(5(5(3(3(0(0(x1)))))))	→	4(3(0(3(3(2(x1))))))	(33)
5(4(4(0(0(4(0(x1)))))))	→	4(2(1(4(5(3(x1))))))	(34)
3(4(4(2(5(2(5(5(5(x1)))))))))	→	2(5(4(0(5(1(0(5(x1))))))))	(35)
0(1(4(1(1(1(0(3(4(0(x1))))))))))	→	3(1(5(0(1(2(2(0(5(2(x1))))))))))	(36)
1(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0(0(0(5(0(3(2(0(3(5(x1))))))))))	(37)
0(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2(2(2(1(3(5(2(1(1(0(x1))))))))))	(38)
0(5(4(2(5(1(2(2(5(4(x1))))))))))	→	0(5(5(5(4(1(4(5(2(4(x1))))))))))	(39)
5(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	0(4(2(3(5(5(5(4(4(2(0(x1)))))))))))	(40)
1(4(4(0(1(4(5(5(3(1(1(2(x1))))))))))))	→	0(2(1(0(4(0(2(4(3(5(5(2(x1))))))))))))	(41)
1(4(5(5(2(3(1(1(1(5(4(4(x1))))))))))))	→	1(3(3(4(0(0(0(2(0(1(2(4(x1))))))))))))	(42)
4(0(5(2(0(2(5(0(1(2(5(4(x1))))))))))))	→	4(0(0(2(0(0(2(3(5(0(0(3(x1))))))))))))	(43)
4(0(4(5(4(1(5(2(4(2(2(5(x1))))))))))))	→	4(1(3(5(2(3(4(5(5(5(4(2(x1))))))))))))	(44)
2(1(5(0(0(5(5(3(4(5(1(4(0(x1)))))))))))))	→	4(2(4(0(2(5(0(5(4(5(3(0(x1))))))))))))	(45)
1(5(1(0(1(5(2(2(0(1(2(2(2(x1)))))))))))))	→	3(3(4(2(1(0(5(2(3(0(4(5(x1))))))))))))	(46)
0(1(2(5(0(5(5(2(2(1(5(5(2(0(x1))))))))))))))	→	1(1(0(4(3(2(1(0(5(5(5(0(2(x1)))))))))))))	(47)
2(1(4(4(2(3(0(0(1(4(3(1(4(2(x1))))))))))))))	→	1(3(4(1(5(4(5(4(4(4(1(5(4(3(x1))))))))))))))	(48)
2(1(5(2(1(5(0(4(3(2(2(0(2(3(3(0(x1))))))))))))))))	→	4(4(3(0(4(2(2(1(1(0(3(1(0(1(0(x1)))))))))))))))	(49)
0(2(5(2(3(4(0(4(2(5(0(2(0(0(2(2(x1))))))))))))))))	→	3(5(2(0(5(2(0(3(1(5(3(1(4(5(4(x1)))))))))))))))	(50)
4(0(4(4(4(0(5(2(1(5(3(5(4(3(4(4(1(x1)))))))))))))))))	→	4(3(5(3(3(3(3(0(1(1(1(0(1(3(3(2(1(x1)))))))))))))))))	(51)
0(2(0(2(0(0(3(4(0(3(3(2(3(0(2(0(3(x1)))))))))))))))))	→	1(1(5(4(0(3(0(2(2(2(1(2(3(5(5(3(x1))))))))))))))))	(52)
0(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	0(5(4(5(1(2(5(1(2(0(2(2(4(5(5(4(4(x1)))))))))))))))))	(53)
3(5(1(5(1(3(1(2(4(1(2(2(4(4(3(4(2(1(x1))))))))))))))))))	→	3(5(3(0(0(0(4(3(0(3(1(0(0(2(3(0(1(x1)))))))))))))))))	(54)
0(1(3(5(5(3(3(4(4(3(3(0(0(4(5(5(5(4(x1))))))))))))))))))	→	1(2(1(4(5(2(1(5(1(4(5(1(1(5(2(3(3(x1)))))))))))))))))	(55)
3(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	3(2(2(1(0(0(5(5(0(3(4(2(1(4(4(2(2(5(3(x1)))))))))))))))))))	(56)
2(3(5(3(3(1(0(3(5(2(2(4(3(0(2(1(3(1(2(0(x1))))))))))))))))))))	→	3(0(3(0(1(1(0(5(1(3(1(1(3(5(2(0(1(3(1(x1)))))))))))))))))))	(57)
2(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	1(2(2(2(0(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))))))))))	(58)
5(0(1(4(2(4(1(5(4(0(5(2(0(0(1(1(3(3(3(0(0(x1)))))))))))))))))))))	→	5(1(1(1(0(5(4(4(1(4(5(3(0(3(2(5(0(3(2(1(x1))))))))))))))))))))	(59)
4(0(3(4(5(2(3(1(1(0(1(1(5(4(1(2(0(1(3(5(0(x1)))))))))))))))))))))	→	4(3(1(4(5(5(3(5(0(0(1(3(5(3(2(4(5(5(1(1(x1))))))))))))))))))))	(60)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

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

all of the following rules can be deleted.

0(1(0(0(x1))))	→	4(3(2(x1)))	(31)
4(5(5(3(3(0(0(x1)))))))	→	4(3(0(3(3(2(x1))))))	(33)
5(4(4(0(0(4(0(x1)))))))	→	4(2(1(4(5(3(x1))))))	(34)
3(4(4(2(5(2(5(5(5(x1)))))))))	→	2(5(4(0(5(1(0(5(x1))))))))	(35)
2(1(5(0(0(5(5(3(4(5(1(4(0(x1)))))))))))))	→	4(2(4(0(2(5(0(5(4(5(3(0(x1))))))))))))	(45)
1(5(1(0(1(5(2(2(0(1(2(2(2(x1)))))))))))))	→	3(3(4(2(1(0(5(2(3(0(4(5(x1))))))))))))	(46)
0(1(2(5(0(5(5(2(2(1(5(5(2(0(x1))))))))))))))	→	1(1(0(4(3(2(1(0(5(5(5(0(2(x1)))))))))))))	(47)
2(1(5(2(1(5(0(4(3(2(2(0(2(3(3(0(x1))))))))))))))))	→	4(4(3(0(4(2(2(1(1(0(3(1(0(1(0(x1)))))))))))))))	(49)
0(2(5(2(3(4(0(4(2(5(0(2(0(0(2(2(x1))))))))))))))))	→	3(5(2(0(5(2(0(3(1(5(3(1(4(5(4(x1)))))))))))))))	(50)
0(2(0(2(0(0(3(4(0(3(3(2(3(0(2(0(3(x1)))))))))))))))))	→	1(1(5(4(0(3(0(2(2(2(1(2(3(5(5(3(x1))))))))))))))))	(52)
3(5(1(5(1(3(1(2(4(1(2(2(4(4(3(4(2(1(x1))))))))))))))))))	→	3(5(3(0(0(0(4(3(0(3(1(0(0(2(3(0(1(x1)))))))))))))))))	(54)
0(1(3(5(5(3(3(4(4(3(3(0(0(4(5(5(5(4(x1))))))))))))))))))	→	1(2(1(4(5(2(1(5(1(4(5(1(1(5(2(3(3(x1)))))))))))))))))	(55)
2(3(5(3(3(1(0(3(5(2(2(4(3(0(2(1(3(1(2(0(x1))))))))))))))))))))	→	3(0(3(0(1(1(0(5(1(3(1(1(3(5(2(0(1(3(1(x1)))))))))))))))))))	(57)
5(0(1(4(2(4(1(5(4(0(5(2(0(0(1(1(3(3(3(0(0(x1)))))))))))))))))))))	→	5(1(1(1(0(5(4(4(1(4(5(3(0(3(2(5(0(3(2(1(x1))))))))))))))))))))	(59)
4(0(3(4(5(2(3(1(1(0(1(1(5(4(1(2(0(1(3(5(0(x1)))))))))))))))))))))	→	4(3(1(4(5(5(3(5(0(0(1(3(5(3(2(4(5(5(1(1(x1))))))))))))))))))))	(60)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 185 ruless (increase limit for explicit display).

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	2^#(2(0(5(2(x1)))))	(71)
2^#(1(4(4(2(3(0(0(1(4(3(1(4(2(x1))))))))))))))	→	3^#(x1)	(174)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(2(1(0(0(5(5(0(3(4(2(1(4(4(2(2(5(3(x1))))))))))))))))))	(208)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(1(0(0(5(5(0(3(4(2(1(4(4(2(2(5(3(x1)))))))))))))))))	(209)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	0^#(0(5(5(0(3(4(2(1(4(4(2(2(5(3(x1)))))))))))))))	(211)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	0^#(1(2(2(0(5(2(x1)))))))	(69)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	2^#(0(5(2(x1))))	(72)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	0^#(5(2(x1)))	(73)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	2^#(x1)	(75)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	2^#(2(2(0(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1)))))))))))))))))))	(227)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	2^#(2(0(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))))))))	(228)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	2^#(0(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1)))))))))))))))))	(229)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))))))	(230)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(2(2(1(3(5(2(1(1(0(x1))))))))))	(86)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(2(1(3(5(2(1(1(0(x1)))))))))	(87)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(1(3(5(2(1(1(0(x1))))))))	(88)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(1(1(0(x1))))	(92)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	1^#(1(0(x1)))	(93)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(0(0(5(0(3(2(0(3(5(x1))))))))))	(76)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	0^#(x1)	(95)
0^#(5(4(2(5(1(2(2(5(4(x1))))))))))	→	0^#(5(5(5(4(1(4(5(2(4(x1))))))))))	(96)
0^#(5(4(2(5(1(2(2(5(4(x1))))))))))	→	5^#(5(5(4(1(4(5(2(4(x1)))))))))	(97)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	0^#(4(2(3(5(5(5(4(4(2(0(x1)))))))))))	(105)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	2^#(3(5(5(5(4(4(2(0(x1)))))))))	(107)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	5^#(5(5(4(4(2(0(x1)))))))	(109)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	5^#(5(4(4(2(0(x1))))))	(110)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	2^#(0(x1))	(114)
0^#(5(4(2(5(1(2(2(5(4(x1))))))))))	→	5^#(5(4(1(4(5(2(4(x1))))))))	(98)
0^#(5(4(2(5(1(2(2(5(4(x1))))))))))	→	2^#(4(x1))	(104)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	0^#(5(4(5(1(2(5(1(2(0(2(2(4(5(5(4(4(x1)))))))))))))))))	(191)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(5(1(2(0(2(2(4(5(5(4(4(x1))))))))))))	(196)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(0(2(2(4(5(5(4(4(x1)))))))))	(199)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	0^#(2(2(4(5(5(4(4(x1))))))))	(200)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(2(4(5(5(4(4(x1)))))))	(201)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(4(5(5(4(4(x1))))))	(202)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	5^#(5(4(4(x1))))	(204)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(0(5(0(3(2(0(3(5(x1)))))))))	(77)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(5(0(3(2(0(3(5(x1))))))))	(78)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(3(2(0(3(5(x1))))))	(80)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	2^#(0(3(5(x1))))	(82)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(3(5(x1)))	(83)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	5^#(x1)	(85)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(4(0(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))))	(232)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))	(234)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(1(3(4(5(4(4(1(1(3(x1))))))))))	(236)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	3^#(x1)	(245)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	0^#(5(5(0(3(4(2(1(4(4(2(2(5(3(x1))))))))))))))	(212)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	5^#(5(0(3(4(2(1(4(4(2(2(5(3(x1)))))))))))))	(213)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	0^#(3(4(2(1(4(4(2(2(5(3(x1)))))))))))	(215)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(1(4(4(2(2(5(3(x1))))))))	(218)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(2(5(3(x1))))	(222)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(5(3(x1)))	(223)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	3^#(x1)	(225)

1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pairs

0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	2^#(2(0(5(2(x1)))))	(71)
2^#(1(4(4(2(3(0(0(1(4(3(1(4(2(x1))))))))))))))	→	3^#(x1)	(174)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(2(1(0(0(5(5(0(3(4(2(1(4(4(2(2(5(3(x1))))))))))))))))))	(208)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(1(0(0(5(5(0(3(4(2(1(4(4(2(2(5(3(x1)))))))))))))))))	(209)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	0^#(0(5(5(0(3(4(2(1(4(4(2(2(5(3(x1)))))))))))))))	(211)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	0^#(1(2(2(0(5(2(x1)))))))	(69)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	2^#(0(5(2(x1))))	(72)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	0^#(5(2(x1)))	(73)
0^#(1(4(1(1(1(0(3(4(0(x1))))))))))	→	2^#(x1)	(75)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	2^#(2(2(0(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1)))))))))))))))))))	(227)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	2^#(2(0(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))))))))	(228)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	2^#(0(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1)))))))))))))))))	(229)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(1(0(4(0(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))))))	(230)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(2(2(1(3(5(2(1(1(0(x1))))))))))	(86)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(2(1(3(5(2(1(1(0(x1)))))))))	(87)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(1(3(5(2(1(1(0(x1))))))))	(88)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	2^#(1(1(0(x1))))	(92)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	1^#(1(0(x1)))	(93)
0^#(0(5(3(0(2(4(1(3(3(x1))))))))))	→	0^#(x1)	(95)
0^#(5(4(2(5(1(2(2(5(4(x1))))))))))	→	5^#(5(5(4(1(4(5(2(4(x1)))))))))	(97)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	2^#(3(5(5(5(4(4(2(0(x1)))))))))	(107)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	5^#(5(5(4(4(2(0(x1)))))))	(109)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	5^#(5(4(4(2(0(x1))))))	(110)
5^#(5(1(4(0(4(4(3(1(1(0(x1)))))))))))	→	2^#(0(x1))	(114)
0^#(5(4(2(5(1(2(2(5(4(x1))))))))))	→	5^#(5(4(1(4(5(2(4(x1))))))))	(98)
0^#(5(4(2(5(1(2(2(5(4(x1))))))))))	→	2^#(4(x1))	(104)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(5(1(2(0(2(2(4(5(5(4(4(x1))))))))))))	(196)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(0(2(2(4(5(5(4(4(x1)))))))))	(199)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	0^#(2(2(4(5(5(4(4(x1))))))))	(200)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(2(4(5(5(4(4(x1)))))))	(201)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	2^#(4(5(5(4(4(x1))))))	(202)
0^#(1(5(4(2(3(5(5(3(1(2(5(3(3(4(0(4(x1)))))))))))))))))	→	5^#(5(4(4(x1))))	(204)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(0(5(0(3(2(0(3(5(x1)))))))))	(77)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(5(0(3(2(0(3(5(x1))))))))	(78)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(3(2(0(3(5(x1))))))	(80)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	2^#(0(3(5(x1))))	(82)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	0^#(3(5(x1)))	(83)
1^#(1(0(3(5(1(2(5(4(0(x1))))))))))	→	5^#(x1)	(85)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(4(0(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))))	(232)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(3(0(1(3(4(5(4(4(1(1(3(x1))))))))))))	(234)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	0^#(1(3(4(5(4(4(1(1(3(x1))))))))))	(236)
2^#(3(4(1(2(0(1(0(5(2(1(1(5(4(0(2(3(0(0(1(x1))))))))))))))))))))	→	3^#(x1)	(245)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	0^#(5(5(0(3(4(2(1(4(4(2(2(5(3(x1))))))))))))))	(212)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	5^#(5(0(3(4(2(1(4(4(2(2(5(3(x1)))))))))))))	(213)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	0^#(3(4(2(1(4(4(2(2(5(3(x1)))))))))))	(215)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(1(4(4(2(2(5(3(x1))))))))	(218)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(2(5(3(x1))))	(222)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	2^#(5(3(x1)))	(223)
3^#(3(2(3(0(2(2(4(1(1(5(0(5(3(0(3(1(1(1(x1)))))))))))))))))))	→	3^#(x1)	(225)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.