Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/86559)

The rewrite relation of the following TRS is considered.

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

9/2

[4(x₁)]

x₁ +

9/2

[3(x₁)]

x₁ +

7/2

[2(x₁)]

x₁ +

9/2

[1(x₁)]

x₁ +

[0(x₁)]

x₁ +

9/2

all of the following rules can be deleted.

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

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 +

[4(x₁)]

x₁ +

[3(x₁)]

x₁ +

[2(x₁)]

x₁ +

[1(x₁)]

x₁ +

[0(x₁)]

x₁ +

[5^#(x₁)]

x₁ +

[4^#(x₁)]

x₁ +

[1^#(x₁)]

x₁ +

[0^#(x₁)]

x₁ +

together with the usable rules

5(0(3(1(4(1(0(1(0(x1)))))))))	→	2(2(4(1(3(2(2(3(0(x1)))))))))	(7)
4(5(1(3(0(0(3(5(2(1(2(5(2(x1)))))))))))))	→	4(5(0(3(5(3(5(5(3(5(4(0(4(x1)))))))))))))	(14)
4(4(5(3(0(2(0(0(0(3(5(0(2(1(x1))))))))))))))	→	2(2(0(4(1(0(0(1(4(2(4(1(3(5(x1))))))))))))))	(17)
0(0(1(1(4(5(3(2(5(3(4(4(3(0(5(4(4(4(2(x1)))))))))))))))))))	→	0(0(4(5(1(4(1(5(3(0(4(0(3(3(4(0(4(0(4(x1)))))))))))))))))))	(26)
1(4(1(4(1(5(2(4(4(3(3(4(5(4(4(1(2(0(2(3(x1))))))))))))))))))))	→	1(2(1(5(0(1(0(4(4(4(2(5(0(1(5(1(1(4(3(3(x1))))))))))))))))))))	(27)

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

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

and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.