Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/4046)

The rewrite relation of the following TRS is considered.

4(2(4(x1)))	→	2(0(0(5(3(3(5(2(0(4(x1))))))))))	(1)
4(4(2(4(2(x1)))))	→	2(0(5(2(1(4(0(2(0(1(x1))))))))))	(2)
0(5(4(2(4(3(x1))))))	→	5(1(5(5(3(5(3(0(0(0(x1))))))))))	(3)
1(1(4(5(3(3(x1))))))	→	1(3(1(1(3(0(1(2(2(1(x1))))))))))	(4)
3(1(4(3(1(2(x1))))))	→	0(0(1(1(4(2(3(0(0(3(x1))))))))))	(5)
3(2(4(2(4(1(x1))))))	→	0(2(1(1(1(5(3(1(3(3(x1))))))))))	(6)
3(3(0(4(1(2(x1))))))	→	3(5(1(2(0(2(0(5(3(1(x1))))))))))	(7)
4(1(4(5(0(5(4(x1)))))))	→	4(1(5(3(1(0(5(3(1(0(x1))))))))))	(8)
4(4(0(5(4(2(2(x1)))))))	→	4(0(4(3(4(4(4(5(4(1(x1))))))))))	(9)
5(4(5(3(2(4(3(x1)))))))	→	2(5(5(5(0(4(5(0(1(4(x1))))))))))	(10)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

4^#(4(2(4(2(x1)))))	→	5^#(2(1(4(0(2(0(1(x1))))))))	(11)
1^#(1(4(5(3(3(x1))))))	→	1^#(2(2(1(x1))))	(12)
5^#(4(5(3(2(4(3(x1)))))))	→	4^#(x1)	(13)
3^#(3(0(4(1(2(x1))))))	→	1^#(2(0(2(0(5(3(1(x1))))))))	(14)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(5(5(0(4(5(0(1(4(x1)))))))))	(15)
1^#(1(4(5(3(3(x1))))))	→	1^#(3(0(1(2(2(1(x1)))))))	(16)
4^#(2(4(x1)))	→	3^#(3(5(2(0(4(x1))))))	(17)
3^#(1(4(3(1(2(x1))))))	→	1^#(1(4(2(3(0(0(3(x1))))))))	(18)
5^#(4(5(3(2(4(3(x1)))))))	→	1^#(4(x1))	(19)
5^#(4(5(3(2(4(3(x1)))))))	→	0^#(1(4(x1)))	(20)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(0(4(5(0(1(4(x1)))))))	(21)
4^#(1(4(5(0(5(4(x1)))))))	→	3^#(1(0(5(3(1(0(x1)))))))	(22)
3^#(1(4(3(1(2(x1))))))	→	1^#(4(2(3(0(0(3(x1)))))))	(23)
0^#(5(4(2(4(3(x1))))))	→	5^#(1(5(5(3(5(3(0(0(0(x1))))))))))	(24)
4^#(1(4(5(0(5(4(x1)))))))	→	1^#(5(3(1(0(5(3(1(0(x1)))))))))	(25)
3^#(1(4(3(1(2(x1))))))	→	4^#(2(3(0(0(3(x1))))))	(26)
4^#(1(4(5(0(5(4(x1)))))))	→	0^#(5(3(1(0(x1)))))	(27)
4^#(1(4(5(0(5(4(x1)))))))	→	1^#(0(5(3(1(0(x1))))))	(28)
0^#(5(4(2(4(3(x1))))))	→	5^#(3(0(0(0(x1)))))	(29)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(x1))	(30)
5^#(4(5(3(2(4(3(x1)))))))	→	0^#(4(5(0(1(4(x1))))))	(31)
4^#(4(2(4(2(x1)))))	→	0^#(1(x1))	(32)
0^#(5(4(2(4(3(x1))))))	→	3^#(0(0(0(x1))))	(33)
0^#(5(4(2(4(3(x1))))))	→	5^#(5(3(5(3(0(0(0(x1))))))))	(34)
4^#(2(4(x1)))	→	3^#(5(2(0(4(x1)))))	(35)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(0(1(4(x1))))	(36)
3^#(1(4(3(1(2(x1))))))	→	0^#(3(x1))	(37)
3^#(2(4(2(4(1(x1))))))	→	3^#(1(3(3(x1))))	(38)
3^#(2(4(2(4(1(x1))))))	→	1^#(1(1(5(3(1(3(3(x1))))))))	(39)
3^#(2(4(2(4(1(x1))))))	→	1^#(5(3(1(3(3(x1))))))	(40)
0^#(5(4(2(4(3(x1))))))	→	3^#(5(3(0(0(0(x1))))))	(41)
4^#(1(4(5(0(5(4(x1)))))))	→	5^#(3(1(0(x1))))	(42)
4^#(4(0(5(4(2(2(x1)))))))	→	5^#(4(1(x1)))	(43)
4^#(1(4(5(0(5(4(x1)))))))	→	4^#(1(5(3(1(0(5(3(1(0(x1))))))))))	(44)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(5(4(1(x1))))	(45)
1^#(1(4(5(3(3(x1))))))	→	3^#(1(1(3(0(1(2(2(1(x1)))))))))	(46)
4^#(1(4(5(0(5(4(x1)))))))	→	1^#(0(x1))	(47)
3^#(1(4(3(1(2(x1))))))	→	0^#(1(1(4(2(3(0(0(3(x1)))))))))	(48)
4^#(2(4(x1)))	→	0^#(4(x1))	(49)
3^#(2(4(2(4(1(x1))))))	→	5^#(3(1(3(3(x1)))))	(50)
3^#(3(0(4(1(2(x1))))))	→	0^#(5(3(1(x1))))	(51)
4^#(1(4(5(0(5(4(x1)))))))	→	0^#(x1)	(52)
0^#(5(4(2(4(3(x1))))))	→	5^#(3(5(3(0(0(0(x1)))))))	(53)
1^#(1(4(5(3(3(x1))))))	→	1^#(1(3(0(1(2(2(1(x1))))))))	(54)
3^#(2(4(2(4(1(x1))))))	→	1^#(3(3(x1)))	(55)
0^#(5(4(2(4(3(x1))))))	→	0^#(x1)	(56)
3^#(3(0(4(1(2(x1))))))	→	0^#(2(0(5(3(1(x1))))))	(57)
3^#(1(4(3(1(2(x1))))))	→	3^#(x1)	(58)
3^#(1(4(3(1(2(x1))))))	→	3^#(0(0(3(x1))))	(59)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(1(x1))	(60)
4^#(4(0(5(4(2(2(x1)))))))	→	3^#(4(4(4(5(4(1(x1)))))))	(61)
4^#(4(2(4(2(x1)))))	→	1^#(x1)	(62)
5^#(4(5(3(2(4(3(x1)))))))	→	4^#(5(0(1(4(x1)))))	(63)
3^#(2(4(2(4(1(x1))))))	→	1^#(1(5(3(1(3(3(x1)))))))	(64)
4^#(2(4(x1)))	→	0^#(5(3(3(5(2(0(4(x1))))))))	(65)
3^#(3(0(4(1(2(x1))))))	→	3^#(1(x1))	(66)
4^#(1(4(5(0(5(4(x1)))))))	→	5^#(3(1(0(5(3(1(0(x1))))))))	(67)
4^#(4(2(4(2(x1)))))	→	0^#(2(0(1(x1))))	(68)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(1(1(4(2(3(0(0(3(x1))))))))))	(69)
4^#(2(4(x1)))	→	5^#(3(3(5(2(0(4(x1)))))))	(70)
3^#(3(0(4(1(2(x1))))))	→	5^#(1(2(0(2(0(5(3(1(x1)))))))))	(71)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(5(4(1(x1)))))	(72)
3^#(3(0(4(1(2(x1))))))	→	1^#(x1)	(73)
1^#(1(4(5(3(3(x1))))))	→	3^#(0(1(2(2(1(x1))))))	(74)
1^#(1(4(5(3(3(x1))))))	→	1^#(x1)	(75)
3^#(3(0(4(1(2(x1))))))	→	3^#(5(1(2(0(2(0(5(3(1(x1))))))))))	(76)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(0(x1)))	(77)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(0(4(3(4(4(4(5(4(1(x1))))))))))	(78)
3^#(3(0(4(1(2(x1))))))	→	5^#(3(1(x1)))	(79)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(3(4(4(4(5(4(1(x1))))))))	(80)
4^#(2(4(x1)))	→	0^#(0(5(3(3(5(2(0(4(x1)))))))))	(81)
4^#(4(0(5(4(2(2(x1)))))))	→	1^#(x1)	(82)
1^#(1(4(5(3(3(x1))))))	→	0^#(1(2(2(1(x1)))))	(83)
4^#(2(4(x1)))	→	5^#(2(0(4(x1))))	(84)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(5(0(4(5(0(1(4(x1))))))))	(85)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(4(5(4(1(x1))))))	(86)
3^#(2(4(2(4(1(x1))))))	→	3^#(x1)	(87)
4^#(4(2(4(2(x1)))))	→	1^#(4(0(2(0(1(x1))))))	(88)
4^#(1(4(5(0(5(4(x1)))))))	→	3^#(1(0(x1)))	(89)
4^#(4(2(4(2(x1)))))	→	0^#(5(2(1(4(0(2(0(1(x1)))))))))	(90)
0^#(5(4(2(4(3(x1))))))	→	1^#(5(5(3(5(3(0(0(0(x1)))))))))	(91)
3^#(2(4(2(4(1(x1))))))	→	0^#(2(1(1(1(5(3(1(3(3(x1))))))))))	(92)
3^#(2(4(2(4(1(x1))))))	→	3^#(3(x1))	(93)
4^#(4(2(4(2(x1)))))	→	4^#(0(2(0(1(x1)))))	(94)
1^#(1(4(5(3(3(x1))))))	→	1^#(3(1(1(3(0(1(2(2(1(x1))))))))))	(95)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(3(x1)))	(96)
4^#(4(0(5(4(2(2(x1)))))))	→	0^#(4(3(4(4(4(5(4(1(x1)))))))))	(97)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

4^#(4(0(5(4(2(2(x1)))))))	→	0^#(4(3(4(4(4(5(4(1(x1)))))))))	(97)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(3(x1)))	(96)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(1(x1))	(60)
3^#(1(4(3(1(2(x1))))))	→	3^#(x1)	(58)
3^#(1(4(3(1(2(x1))))))	→	3^#(0(0(3(x1))))	(59)
0^#(5(4(2(4(3(x1))))))	→	0^#(x1)	(56)
3^#(2(4(2(4(1(x1))))))	→	3^#(3(x1))	(93)
4^#(1(4(5(0(5(4(x1)))))))	→	0^#(x1)	(52)
0^#(5(4(2(4(3(x1))))))	→	5^#(3(5(3(0(0(0(x1)))))))	(53)
3^#(3(0(4(1(2(x1))))))	→	0^#(5(3(1(x1))))	(51)
3^#(2(4(2(4(1(x1))))))	→	5^#(3(1(3(3(x1)))))	(50)
4^#(2(4(x1)))	→	0^#(4(x1))	(49)
3^#(1(4(3(1(2(x1))))))	→	0^#(1(1(4(2(3(0(0(3(x1)))))))))	(48)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(5(4(1(x1))))	(45)
4^#(1(4(5(0(5(4(x1)))))))	→	3^#(1(0(x1)))	(89)
4^#(1(4(5(0(5(4(x1)))))))	→	4^#(1(5(3(1(0(5(3(1(0(x1))))))))))	(44)
4^#(4(0(5(4(2(2(x1)))))))	→	5^#(4(1(x1)))	(43)
3^#(2(4(2(4(1(x1))))))	→	3^#(x1)	(87)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(4(5(4(1(x1))))))	(86)
4^#(1(4(5(0(5(4(x1)))))))	→	5^#(3(1(0(x1))))	(42)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(5(0(4(5(0(1(4(x1))))))))	(85)
0^#(5(4(2(4(3(x1))))))	→	3^#(5(3(0(0(0(x1))))))	(41)
3^#(2(4(2(4(1(x1))))))	→	3^#(1(3(3(x1))))	(38)
3^#(1(4(3(1(2(x1))))))	→	0^#(3(x1))	(37)
3^#(3(0(4(1(2(x1))))))	→	5^#(3(1(x1)))	(79)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(3(4(4(4(5(4(1(x1))))))))	(80)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(0(4(3(4(4(4(5(4(1(x1))))))))))	(78)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(0(1(4(x1))))	(36)
0^#(5(4(2(4(3(x1))))))	→	5^#(5(3(5(3(0(0(0(x1))))))))	(34)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(0(x1)))	(77)
0^#(5(4(2(4(3(x1))))))	→	3^#(0(0(0(x1))))	(33)
4^#(4(2(4(2(x1)))))	→	0^#(1(x1))	(32)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(x1))	(30)
5^#(4(5(3(2(4(3(x1)))))))	→	0^#(4(5(0(1(4(x1))))))	(31)
0^#(5(4(2(4(3(x1))))))	→	5^#(3(0(0(0(x1)))))	(29)
4^#(1(4(5(0(5(4(x1)))))))	→	0^#(5(3(1(0(x1)))))	(27)
3^#(1(4(3(1(2(x1))))))	→	4^#(2(3(0(0(3(x1))))))	(26)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(5(4(1(x1)))))	(72)
0^#(5(4(2(4(3(x1))))))	→	5^#(1(5(5(3(5(3(0(0(0(x1))))))))))	(24)
4^#(1(4(5(0(5(4(x1)))))))	→	3^#(1(0(5(3(1(0(x1)))))))	(22)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(1(1(4(2(3(0(0(3(x1))))))))))	(69)
4^#(1(4(5(0(5(4(x1)))))))	→	5^#(3(1(0(5(3(1(0(x1))))))))	(67)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(0(4(5(0(1(4(x1)))))))	(21)
3^#(3(0(4(1(2(x1))))))	→	3^#(1(x1))	(66)
5^#(4(5(3(2(4(3(x1)))))))	→	0^#(1(4(x1)))	(20)
5^#(4(5(3(2(4(3(x1)))))))	→	4^#(5(0(1(4(x1)))))	(63)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(5(5(0(4(5(0(1(4(x1)))))))))	(15)
5^#(4(5(3(2(4(3(x1)))))))	→	4^#(x1)	(13)
4^#(4(0(5(4(2(2(x1)))))))	→	3^#(4(4(4(5(4(1(x1)))))))	(61)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

1(1(4(5(3(3(x1))))))	→	1(3(1(1(3(0(1(2(2(1(x1))))))))))	(4)
4(1(4(5(0(5(4(x1)))))))	→	4(1(5(3(1(0(5(3(1(0(x1))))))))))	(8)
4(2(4(x1)))	→	2(0(0(5(3(3(5(2(0(4(x1))))))))))	(1)
0(5(4(2(4(3(x1))))))	→	5(1(5(5(3(5(3(0(0(0(x1))))))))))	(3)
3(1(4(3(1(2(x1))))))	→	0(0(1(1(4(2(3(0(0(3(x1))))))))))	(5)
5(4(5(3(2(4(3(x1)))))))	→	2(5(5(5(0(4(5(0(1(4(x1))))))))))	(10)
3(3(0(4(1(2(x1))))))	→	3(5(1(2(0(2(0(5(3(1(x1))))))))))	(7)
4(4(0(5(4(2(2(x1)))))))	→	4(0(4(3(4(4(4(5(4(1(x1))))))))))	(9)
3(2(4(2(4(1(x1))))))	→	0(2(1(1(1(5(3(1(3(3(x1))))))))))	(6)
4(4(2(4(2(x1)))))	→	2(0(5(2(1(4(0(2(0(1(x1))))))))))	(2)

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

0^#(5(4(2(4(3(x1))))))	→	5^#(3(5(3(0(0(0(x1)))))))	(53)
3^#(2(4(2(4(1(x1))))))	→	5^#(3(1(3(3(x1)))))	(50)
4^#(4(0(5(4(2(2(x1)))))))	→	5^#(4(1(x1)))	(43)
4^#(1(4(5(0(5(4(x1)))))))	→	5^#(3(1(0(x1))))	(42)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(5(0(4(5(0(1(4(x1))))))))	(85)
3^#(3(0(4(1(2(x1))))))	→	5^#(3(1(x1)))	(79)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(0(1(4(x1))))	(36)
0^#(5(4(2(4(3(x1))))))	→	5^#(5(3(5(3(0(0(0(x1))))))))	(34)
5^#(4(5(3(2(4(3(x1)))))))	→	0^#(4(5(0(1(4(x1))))))	(31)
0^#(5(4(2(4(3(x1))))))	→	5^#(3(0(0(0(x1)))))	(29)
0^#(5(4(2(4(3(x1))))))	→	5^#(1(5(5(3(5(3(0(0(0(x1))))))))))	(24)
4^#(1(4(5(0(5(4(x1)))))))	→	5^#(3(1(0(5(3(1(0(x1))))))))	(67)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(0(4(5(0(1(4(x1)))))))	(21)
5^#(4(5(3(2(4(3(x1)))))))	→	0^#(1(4(x1)))	(20)
5^#(4(5(3(2(4(3(x1)))))))	→	4^#(5(0(1(4(x1)))))	(63)
5^#(4(5(3(2(4(3(x1)))))))	→	5^#(5(5(0(4(5(0(1(4(x1)))))))))	(15)
5^#(4(5(3(2(4(3(x1)))))))	→	4^#(x1)	(13)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

4^#(1(4(5(0(5(4(x1)))))))	→	4^#(1(5(3(1(0(5(3(1(0(x1))))))))))	(44)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(1(x1))	(60)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(5(4(1(x1))))	(45)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(5(4(1(x1)))))	(72)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(4(5(4(1(x1))))))	(86)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(3(4(4(4(5(4(1(x1))))))))	(80)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(0(4(3(4(4(4(5(4(1(x1))))))))))	(78)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	5
[1(x₁)]	=	1445
[4(x₁)]	=	x₁ + 1
[5(x₁)]	=	19354
[3(x₁)]	=	19356
[4^#(x₁)]	=	x₁ + 5
[0(x₁)]	=	19356
[3^#(x₁)]	=	5
[5^#(x₁)]	=	x₁ + 0
[2(x₁)]	=	841
[1^#(x₁)]	=	0

together with the usable rules

1(1(4(5(3(3(x1))))))	→	1(3(1(1(3(0(1(2(2(1(x1))))))))))	(4)
4(1(4(5(0(5(4(x1)))))))	→	4(1(5(3(1(0(5(3(1(0(x1))))))))))	(8)
4(2(4(x1)))	→	2(0(0(5(3(3(5(2(0(4(x1))))))))))	(1)
0(5(4(2(4(3(x1))))))	→	5(1(5(5(3(5(3(0(0(0(x1))))))))))	(3)
3(1(4(3(1(2(x1))))))	→	0(0(1(1(4(2(3(0(0(3(x1))))))))))	(5)
5(4(5(3(2(4(3(x1)))))))	→	2(5(5(5(0(4(5(0(1(4(x1))))))))))	(10)
3(3(0(4(1(2(x1))))))	→	3(5(1(2(0(2(0(5(3(1(x1))))))))))	(7)
4(4(0(5(4(2(2(x1)))))))	→	4(0(4(3(4(4(4(5(4(1(x1))))))))))	(9)
3(2(4(2(4(1(x1))))))	→	0(2(1(1(1(5(3(1(3(3(x1))))))))))	(6)
4(4(2(4(2(x1)))))	→	2(0(5(2(1(4(0(2(0(1(x1))))))))))	(2)

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

4^#(4(0(5(4(2(2(x1)))))))	→	4^#(1(x1))	(60)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(5(4(1(x1))))	(45)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(5(4(1(x1)))))	(72)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(4(4(5(4(1(x1))))))	(86)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(3(4(4(4(5(4(1(x1))))))))	(80)
4^#(4(0(5(4(2(2(x1)))))))	→	4^#(0(4(3(4(4(4(5(4(1(x1))))))))))	(78)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

4^#(1(4(5(0(5(4(x1)))))))

→

4^#(1(5(3(1(0(5(3(1(0(x1))))))))))

(44)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	5
[1(x₁)]	=	x₁ + 12213
[4(x₁)]	=	x₁ + 1
[5(x₁)]	=	1
[3(x₁)]	=	2
[4^#(x₁)]	=	x₁ + 5
[0(x₁)]	=	1
[3^#(x₁)]	=	5
[5^#(x₁)]	=	x₁ + 0
[2(x₁)]	=	1
[1^#(x₁)]	=	0

together with the usable rules

1(1(4(5(3(3(x1))))))	→	1(3(1(1(3(0(1(2(2(1(x1))))))))))	(4)
4(1(4(5(0(5(4(x1)))))))	→	4(1(5(3(1(0(5(3(1(0(x1))))))))))	(8)
4(2(4(x1)))	→	2(0(0(5(3(3(5(2(0(4(x1))))))))))	(1)
0(5(4(2(4(3(x1))))))	→	5(1(5(5(3(5(3(0(0(0(x1))))))))))	(3)
3(1(4(3(1(2(x1))))))	→	0(0(1(1(4(2(3(0(0(3(x1))))))))))	(5)
5(4(5(3(2(4(3(x1)))))))	→	2(5(5(5(0(4(5(0(1(4(x1))))))))))	(10)
3(3(0(4(1(2(x1))))))	→	3(5(1(2(0(2(0(5(3(1(x1))))))))))	(7)
4(4(0(5(4(2(2(x1)))))))	→	4(0(4(3(4(4(4(5(4(1(x1))))))))))	(9)
3(2(4(2(4(1(x1))))))	→	0(2(1(1(1(5(3(1(3(3(x1))))))))))	(6)
4(4(2(4(2(x1)))))	→	2(0(5(2(1(4(0(2(0(1(x1))))))))))	(2)

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

4^#(1(4(5(0(5(4(x1)))))))

→

4^#(1(5(3(1(0(5(3(1(0(x1))))))))))

(44)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

4^#(2(4(x1)))	→	0^#(4(x1))	(49)
0^#(5(4(2(4(3(x1))))))	→	0^#(x1)	(56)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(x1))	(30)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(0(x1)))	(77)
0^#(5(4(2(4(3(x1))))))	→	3^#(0(0(0(x1))))	(33)
0^#(5(4(2(4(3(x1))))))	→	3^#(5(3(0(0(0(x1))))))	(41)
3^#(1(4(3(1(2(x1))))))	→	3^#(x1)	(58)
3^#(1(4(3(1(2(x1))))))	→	0^#(3(x1))	(37)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(3(x1)))	(96)
3^#(1(4(3(1(2(x1))))))	→	3^#(0(0(3(x1))))	(59)
3^#(1(4(3(1(2(x1))))))	→	4^#(2(3(0(0(3(x1))))))	(26)
3^#(1(4(3(1(2(x1))))))	→	0^#(1(1(4(2(3(0(0(3(x1)))))))))	(48)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(1(1(4(2(3(0(0(3(x1))))))))))	(69)
3^#(3(0(4(1(2(x1))))))	→	3^#(1(x1))	(66)
3^#(3(0(4(1(2(x1))))))	→	0^#(5(3(1(x1))))	(51)
3^#(2(4(2(4(1(x1))))))	→	3^#(x1)	(87)
3^#(2(4(2(4(1(x1))))))	→	3^#(3(x1))	(93)
3^#(2(4(2(4(1(x1))))))	→	3^#(1(3(3(x1))))	(38)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

1(1(4(5(3(3(x1))))))	→	1(3(1(1(3(0(1(2(2(1(x1))))))))))	(4)
4(1(4(5(0(5(4(x1)))))))	→	4(1(5(3(1(0(5(3(1(0(x1))))))))))	(8)
4(2(4(x1)))	→	2(0(0(5(3(3(5(2(0(4(x1))))))))))	(1)
0(5(4(2(4(3(x1))))))	→	5(1(5(5(3(5(3(0(0(0(x1))))))))))	(3)
3(1(4(3(1(2(x1))))))	→	0(0(1(1(4(2(3(0(0(3(x1))))))))))	(5)
5(4(5(3(2(4(3(x1)))))))	→	2(5(5(5(0(4(5(0(1(4(x1))))))))))	(10)
3(3(0(4(1(2(x1))))))	→	3(5(1(2(0(2(0(5(3(1(x1))))))))))	(7)
4(4(0(5(4(2(2(x1)))))))	→	4(0(4(3(4(4(4(5(4(1(x1))))))))))	(9)
3(2(4(2(4(1(x1))))))	→	0(2(1(1(1(5(3(1(3(3(x1))))))))))	(6)
4(4(2(4(2(x1)))))	→	2(0(5(2(1(4(0(2(0(1(x1))))))))))	(2)

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

4^#(2(4(x1)))	→	0^#(4(x1))	(49)
3^#(1(4(3(1(2(x1))))))	→	4^#(2(3(0(0(3(x1))))))	(26)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(5(4(2(4(3(x1))))))	→	0^#(x1)	(56)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(x1))	(30)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(0(x1)))	(77)
0^#(5(4(2(4(3(x1))))))	→	3^#(0(0(0(x1))))	(33)
0^#(5(4(2(4(3(x1))))))	→	3^#(5(3(0(0(0(x1))))))	(41)
3^#(1(4(3(1(2(x1))))))	→	3^#(x1)	(58)
3^#(1(4(3(1(2(x1))))))	→	0^#(3(x1))	(37)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(3(x1)))	(96)
3^#(1(4(3(1(2(x1))))))	→	3^#(0(0(3(x1))))	(59)
3^#(1(4(3(1(2(x1))))))	→	0^#(1(1(4(2(3(0(0(3(x1)))))))))	(48)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(1(1(4(2(3(0(0(3(x1))))))))))	(69)
3^#(3(0(4(1(2(x1))))))	→	3^#(1(x1))	(66)
3^#(3(0(4(1(2(x1))))))	→	0^#(5(3(1(x1))))	(51)
3^#(2(4(2(4(1(x1))))))	→	3^#(x1)	(87)
3^#(2(4(2(4(1(x1))))))	→	3^#(3(x1))	(93)
3^#(2(4(2(4(1(x1))))))	→	3^#(1(3(3(x1))))	(38)

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[0^#(x₁)]

0	1
0	1

· x₁ +

[1(x₁)]

[4(x₁)]

1	1
1	0

· x₁ +

[5(x₁)]

0	1
1	0

· x₁ +

[3(x₁)]

0	1
0	0

· x₁ +

[4^#(x₁)]

[0(x₁)]

0	0
0	1

· x₁ +

[3^#(x₁)]

[5^#(x₁)]

[2(x₁)]

0	1
0	1

· x₁ +

[1^#(x₁)]

together with the usable rules

1(1(4(5(3(3(x1))))))	→	1(3(1(1(3(0(1(2(2(1(x1))))))))))	(4)
4(1(4(5(0(5(4(x1)))))))	→	4(1(5(3(1(0(5(3(1(0(x1))))))))))	(8)
4(2(4(x1)))	→	2(0(0(5(3(3(5(2(0(4(x1))))))))))	(1)
0(5(4(2(4(3(x1))))))	→	5(1(5(5(3(5(3(0(0(0(x1))))))))))	(3)
3(1(4(3(1(2(x1))))))	→	0(0(1(1(4(2(3(0(0(3(x1))))))))))	(5)
5(4(5(3(2(4(3(x1)))))))	→	2(5(5(5(0(4(5(0(1(4(x1))))))))))	(10)
3(3(0(4(1(2(x1))))))	→	3(5(1(2(0(2(0(5(3(1(x1))))))))))	(7)
4(4(0(5(4(2(2(x1)))))))	→	4(0(4(3(4(4(4(5(4(1(x1))))))))))	(9)
3(2(4(2(4(1(x1))))))	→	0(2(1(1(1(5(3(1(3(3(x1))))))))))	(6)
4(4(2(4(2(x1)))))	→	2(0(5(2(1(4(0(2(0(1(x1))))))))))	(2)

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

0^#(5(4(2(4(3(x1))))))	→	0^#(x1)	(56)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(x1))	(30)
0^#(5(4(2(4(3(x1))))))	→	0^#(0(0(x1)))	(77)
0^#(5(4(2(4(3(x1))))))	→	3^#(0(0(0(x1))))	(33)
0^#(5(4(2(4(3(x1))))))	→	3^#(5(3(0(0(0(x1))))))	(41)
3^#(1(4(3(1(2(x1))))))	→	0^#(3(x1))	(37)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(3(x1)))	(96)
3^#(1(4(3(1(2(x1))))))	→	0^#(1(1(4(2(3(0(0(3(x1)))))))))	(48)
3^#(1(4(3(1(2(x1))))))	→	0^#(0(1(1(4(2(3(0(0(3(x1))))))))))	(69)
3^#(3(0(4(1(2(x1))))))	→	0^#(5(3(1(x1))))	(51)

could be deleted.

1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

3^#(1(4(3(1(2(x1))))))	→	3^#(x1)	(58)
3^#(1(4(3(1(2(x1))))))	→	3^#(0(0(3(x1))))	(59)
3^#(3(0(4(1(2(x1))))))	→	3^#(1(x1))	(66)
3^#(2(4(2(4(1(x1))))))	→	3^#(x1)	(87)
3^#(2(4(2(4(1(x1))))))	→	3^#(3(x1))	(93)
3^#(2(4(2(4(1(x1))))))	→	3^#(1(3(3(x1))))	(38)

1.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[0^#(x₁)]

[1(x₁)]

0	1
1	0

· x₁ +

[4(x₁)]

1	1
1	0

· x₁ +

66071

33035

[5(x₁)]

[3(x₁)]

0	0
1	0

· x₁ +

33036

[4^#(x₁)]

[0(x₁)]

0	1
0	0

· x₁ +

33030

[3^#(x₁)]

0	1
0	0

· x₁ +

[5^#(x₁)]

[2(x₁)]

0	0
1	1

· x₁ +

[1^#(x₁)]

together with the usable rules

1(1(4(5(3(3(x1))))))	→	1(3(1(1(3(0(1(2(2(1(x1))))))))))	(4)
4(1(4(5(0(5(4(x1)))))))	→	4(1(5(3(1(0(5(3(1(0(x1))))))))))	(8)
4(2(4(x1)))	→	2(0(0(5(3(3(5(2(0(4(x1))))))))))	(1)
0(5(4(2(4(3(x1))))))	→	5(1(5(5(3(5(3(0(0(0(x1))))))))))	(3)
3(1(4(3(1(2(x1))))))	→	0(0(1(1(4(2(3(0(0(3(x1))))))))))	(5)
5(4(5(3(2(4(3(x1)))))))	→	2(5(5(5(0(4(5(0(1(4(x1))))))))))	(10)
3(3(0(4(1(2(x1))))))	→	3(5(1(2(0(2(0(5(3(1(x1))))))))))	(7)
4(4(0(5(4(2(2(x1)))))))	→	4(0(4(3(4(4(4(5(4(1(x1))))))))))	(9)
3(2(4(2(4(1(x1))))))	→	0(2(1(1(1(5(3(1(3(3(x1))))))))))	(6)
4(4(2(4(2(x1)))))	→	2(0(5(2(1(4(0(2(0(1(x1))))))))))	(2)

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

3^#(1(4(3(1(2(x1))))))	→	3^#(x1)	(58)
3^#(1(4(3(1(2(x1))))))	→	3^#(0(0(3(x1))))	(59)
3^#(3(0(4(1(2(x1))))))	→	3^#(1(x1))	(66)
3^#(2(4(2(4(1(x1))))))	→	3^#(x1)	(87)
3^#(2(4(2(4(1(x1))))))	→	3^#(3(x1))	(93)
3^#(2(4(2(4(1(x1))))))	→	3^#(1(3(3(x1))))	(38)

could be deleted.

1.1.1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

1^#(1(4(5(3(3(x1))))))

→

1^#(x1)

(75)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	5
[1(x₁)]	=	x₁ + 7362
[4(x₁)]	=	x₁ + 8573
[5(x₁)]	=	x₁ + 33267
[3(x₁)]	=	x₁ + 40628
[4^#(x₁)]	=	3
[0(x₁)]	=	104554
[3^#(x₁)]	=	7
[5^#(x₁)]	=	0
[2(x₁)]	=	x₁ + 0
[1^#(x₁)]	=	x₁ + 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

1^#(1(4(5(3(3(x1))))))

→

1^#(x1)

(75)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.