Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/85834)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[5(x₁)]

x₁ +

[4(x₁)]

x₁ +

12/17

[3(x₁)]

x₁ +

13/17

[2(x₁)]

x₁ +

13/17

[1(x₁)]

x₁ +

13/17

[0(x₁)]

x₁ +

all of the following rules can be deleted.

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	5^#(3(1(3(3(x1)))))	(31)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	5^#(0(2(0(3(0(2(5(3(1(3(3(x1))))))))))))	(32)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	4^#(5(0(2(0(3(0(2(5(3(1(3(3(x1)))))))))))))	(33)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	3^#(1(3(3(x1))))	(34)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	3^#(0(2(5(3(1(3(3(x1))))))))	(35)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	2^#(5(3(1(3(3(x1))))))	(36)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	2^#(4(5(0(2(0(3(0(2(5(3(1(3(3(x1))))))))))))))	(37)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	2^#(0(3(0(2(5(3(1(3(3(x1))))))))))	(38)
4^#(4(4(3(5(1(4(0(x1))))))))	→	3^#(2(x1))	(39)
4^#(4(4(3(5(1(4(0(x1))))))))	→	3^#(0(2(2(2(2(3(2(x1))))))))	(40)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(x1)	(41)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(3(2(x1)))	(42)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(2(3(2(x1))))	(43)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(2(2(3(2(x1)))))	(44)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(2(2(2(3(2(x1))))))	(45)
4^#(3(2(5(0(x1)))))	→	5^#(x1)	(46)
4^#(3(2(5(0(x1)))))	→	5^#(5(x1))	(47)
4^#(3(2(5(0(x1)))))	→	4^#(1(5(5(x1))))	(48)
4^#(3(2(5(0(x1)))))	→	3^#(4(1(5(5(x1)))))	(49)
4^#(0(5(3(5(1(3(5(x1))))))))	→	4^#(0(0(1(3(0(1(5(x1))))))))	(50)
4^#(0(5(3(5(1(3(5(x1))))))))	→	3^#(0(1(5(x1))))	(51)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(5(5(4(0(4(1(2(5(5(0(x1)))))))))))	(52)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(5(4(0(4(1(2(5(5(0(x1))))))))))	(53)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(5(0(x1)))	(54)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(4(0(4(1(2(5(5(0(x1)))))))))	(55)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))))	(56)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(0(x1))	(57)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))))))	(58)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	4^#(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))))))	(59)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	4^#(1(2(5(5(0(x1))))))	(60)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	4^#(0(4(1(2(5(5(0(x1))))))))	(61)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	3^#(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))	(62)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	3^#(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))))))))	(63)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	3^#(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))	(64)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	2^#(5(5(0(x1))))	(65)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(4(2(4(1(4(0(5(x1))))))))	(66)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(3(0(4(4(2(4(1(4(0(5(x1)))))))))))	(67)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(2(4(1(4(0(5(x1)))))))	(68)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(1(4(0(5(x1)))))	(69)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(0(5(x1)))	(70)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	3^#(0(4(4(2(4(1(4(0(5(x1))))))))))	(71)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	2^#(4(1(4(0(5(x1))))))	(72)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	5^#(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))))))))	(73)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	5^#(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))))))	(74)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))))))	(75)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(4(2(1(2(0(1(0(x1))))))))	(76)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))	(77)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(2(1(2(0(1(0(x1)))))))	(78)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))))	(79)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))	(80)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))	(81)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(1(1(4(4(2(1(2(0(1(0(x1)))))))))))	(82)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))	(83)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	2^#(1(2(0(1(0(x1))))))	(84)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	2^#(0(1(0(x1))))	(85)

1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[5(x₁)]

x₁ +

13/2

[4(x₁)]

x₁ +

78/17

[3(x₁)]

x₁ +

169/34

[2(x₁)]

x₁ +

169/34

[1(x₁)]

x₁ +

169/34

[0(x₁)]

x₁ +

13/2

[5^#(x₁)]

x₁ +

39/17

[4^#(x₁)]

x₁ +

83/17

[3^#(x₁)]

x₁ +

143/34

[2^#(x₁)]

x₁ +

13/17

together with the usable rules

4(3(2(5(0(x1)))))	→	3(4(1(5(5(x1)))))	(5)
4(0(5(3(5(1(3(5(x1))))))))	→	4(0(0(1(3(0(1(5(x1))))))))	(10)
4(4(4(3(5(1(4(0(x1))))))))	→	3(0(2(2(2(2(3(2(x1))))))))	(11)
3(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	0(4(3(0(4(4(2(4(1(4(0(5(x1))))))))))))	(13)
5(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	2(4(5(0(2(0(3(0(2(5(3(1(3(3(x1))))))))))))))	(18)
3(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	3(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))))))))	(29)
2(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	5(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))))))))	(30)

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

5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	5^#(3(1(3(3(x1)))))	(31)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	5^#(0(2(0(3(0(2(5(3(1(3(3(x1))))))))))))	(32)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	4^#(5(0(2(0(3(0(2(5(3(1(3(3(x1)))))))))))))	(33)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	3^#(1(3(3(x1))))	(34)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	3^#(0(2(5(3(1(3(3(x1))))))))	(35)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	2^#(5(3(1(3(3(x1))))))	(36)
5^#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1))))))))))))))	→	2^#(0(3(0(2(5(3(1(3(3(x1))))))))))	(38)
4^#(4(4(3(5(1(4(0(x1))))))))	→	3^#(2(x1))	(39)
4^#(4(4(3(5(1(4(0(x1))))))))	→	3^#(0(2(2(2(2(3(2(x1))))))))	(40)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(x1)	(41)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(3(2(x1)))	(42)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(2(3(2(x1))))	(43)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(2(2(3(2(x1)))))	(44)
4^#(4(4(3(5(1(4(0(x1))))))))	→	2^#(2(2(2(3(2(x1))))))	(45)
4^#(3(2(5(0(x1)))))	→	5^#(x1)	(46)
4^#(3(2(5(0(x1)))))	→	5^#(5(x1))	(47)
4^#(3(2(5(0(x1)))))	→	4^#(1(5(5(x1))))	(48)
4^#(3(2(5(0(x1)))))	→	3^#(4(1(5(5(x1)))))	(49)
4^#(0(5(3(5(1(3(5(x1))))))))	→	3^#(0(1(5(x1))))	(51)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(5(5(4(0(4(1(2(5(5(0(x1)))))))))))	(52)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(5(4(0(4(1(2(5(5(0(x1))))))))))	(53)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(5(0(x1)))	(54)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(4(0(4(1(2(5(5(0(x1)))))))))	(55)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))))	(56)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(0(x1))	(57)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	5^#(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))))))	(58)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	4^#(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))))))	(59)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	4^#(1(2(5(5(0(x1))))))	(60)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	4^#(0(4(1(2(5(5(0(x1))))))))	(61)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	3^#(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))	(62)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	3^#(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))	(64)
3^#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1))))))))))))))))))))	→	2^#(5(5(0(x1))))	(65)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(4(2(4(1(4(0(5(x1))))))))	(66)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(3(0(4(4(2(4(1(4(0(5(x1)))))))))))	(67)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(2(4(1(4(0(5(x1)))))))	(68)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(1(4(0(5(x1)))))	(69)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	4^#(0(5(x1)))	(70)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	3^#(0(4(4(2(4(1(4(0(5(x1))))))))))	(71)
3^#(2(2(0(0(0(0(3(1(0(5(x1)))))))))))	→	2^#(4(1(4(0(5(x1))))))	(72)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	5^#(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))))))	(74)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))))))	(75)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(4(2(1(2(0(1(0(x1))))))))	(76)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))	(77)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(2(1(2(0(1(0(x1)))))))	(78)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	4^#(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))))	(79)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))	(80)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))	(81)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(1(1(4(4(2(1(2(0(1(0(x1)))))))))))	(82)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	3^#(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))	(83)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	2^#(1(2(0(1(0(x1))))))	(84)
2^#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1)))))))))))))))))))))	→	2^#(0(1(0(x1))))	(85)

and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.