Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/86745)

The rewrite relation of the following TRS is considered.

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

667/13

[3(x₁)]

x₁ +

803/13

[2(x₁)]

x₁ +

[1(x₁)]

x₁ +

730/13

[0(x₁)]

x₁ +

all of the following rules can be deleted.

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

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 +

39/44

[3(x₁)]

x₁ +

[2(x₁)]

x₁ +

41/44

[1(x₁)]

x₁ +

10/11

[0(x₁)]

x₁ +

15/22

[5^#(x₁)]

x₁ +

[4^#(x₁)]

x₁ +

[3^#(x₁)]

x₁ +

[0^#(x₁)]

x₁ +

together with the usable rules

0(4(3(3(4(4(1(x1)))))))	→	4(4(2(5(0(2(2(x1)))))))	(6)
5(1(0(5(2(2(2(3(3(2(5(1(5(1(x1))))))))))))))	→	5(4(0(1(4(3(2(2(3(3(3(5(5(1(x1))))))))))))))	(17)
3(2(1(2(4(2(1(1(3(3(3(5(2(2(0(4(4(x1)))))))))))))))))	→	3(2(3(1(0(2(0(2(4(5(5(4(1(0(0(2(5(0(x1))))))))))))))))))	(22)
4(5(2(3(5(4(5(0(5(1(2(3(0(1(1(0(3(5(0(3(0(x1)))))))))))))))))))))	→	4(4(3(1(4(4(0(5(3(5(2(1(4(2(4(1(0(2(4(5(0(x1)))))))))))))))))))))	(29)
5(1(4(0(0(3(4(2(3(0(3(5(4(0(4(2(4(0(0(5(0(x1)))))))))))))))))))))	→	5(0(3(2(2(0(4(1(1(5(3(0(1(5(0(1(3(2(2(3(x1))))))))))))))))))))	(30)

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

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

and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.