Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/85079)

The rewrite relation of the following TRS is considered.

0(1(2(2(3(x1)))))	→	0(4(1(4(3(x1)))))	(1)
2(1(4(3(1(x1)))))	→	4(1(5(3(x1))))	(2)
1(2(3(5(3(0(x1))))))	→	1(1(4(3(5(1(x1))))))	(3)
2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
0(0(1(2(2(0(4(2(x1))))))))	→	2(4(0(0(4(4(3(0(x1))))))))	(6)
0(1(1(3(2(3(1(3(2(x1)))))))))	→	0(0(5(3(3(1(1(0(x1))))))))	(7)
1(2(5(5(5(2(5(3(5(x1)))))))))	→	4(5(3(2(5(1(3(2(5(x1)))))))))	(8)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
0(1(1(1(2(2(4(1(1(1(x1))))))))))	→	2(1(2(4(0(2(2(5(0(x1)))))))))	(10)
2(2(2(1(4(1(1(3(0(2(x1))))))))))	→	0(2(0(4(2(3(5(4(0(x1)))))))))	(11)
2(2(4(2(1(2(3(2(1(4(x1))))))))))	→	3(3(3(0(4(5(0(2(4(x1)))))))))	(12)
1(3(1(5(0(4(2(2(0(5(5(x1)))))))))))	→	4(4(1(2(2(5(2(2(2(0(5(x1)))))))))))	(13)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
3(2(1(1(4(2(4(5(2(5(3(3(x1))))))))))))	→	5(1(3(1(2(4(1(2(2(1(0(x1)))))))))))	(16)
5(1(3(3(0(3(5(2(1(4(1(5(x1))))))))))))	→	3(5(5(5(5(0(1(5(4(2(5(x1)))))))))))	(17)
5(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))	→	5(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))	(18)
0(1(3(1(5(2(2(4(4(5(5(2(4(x1)))))))))))))	→	0(3(4(0(0(4(0(2(1(3(4(3(4(x1)))))))))))))	(19)
1(2(5(2(4(4(0(3(4(0(3(3(2(x1)))))))))))))	→	1(2(4(2(5(3(5(2(1(3(2(0(x1))))))))))))	(20)
5(5(0(2(4(4(2(4(2(2(0(3(4(x1)))))))))))))	→	5(3(5(2(0(0(4(4(5(5(5(2(x1))))))))))))	(21)
1(3(5(3(2(3(4(1(5(1(4(1(0(3(x1))))))))))))))	→	1(1(1(0(0(2(0(3(4(2(1(3(0(3(x1))))))))))))))	(22)
2(2(4(0(2(2(4(3(5(0(2(3(3(5(3(x1)))))))))))))))	→	2(1(5(1(3(5(5(5(5(0(0(2(5(x1)))))))))))))	(23)
5(0(2(2(0(2(4(3(2(2(4(5(0(2(3(x1)))))))))))))))	→	5(4(1(1(5(4(5(0(2(3(0(3(3(3(x1))))))))))))))	(24)
0(1(4(2(0(5(1(5(2(4(0(0(3(4(4(2(x1))))))))))))))))	→	4(1(5(4(1(0(2(0(3(4(4(1(1(3(4(2(x1))))))))))))))))	(25)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
3(3(3(4(0(3(1(4(4(4(4(1(3(3(1(4(x1))))))))))))))))	→	5(5(2(4(0(0(3(0(4(5(0(2(2(2(x1))))))))))))))	(27)
3(3(3(5(3(1(1(2(2(3(4(2(0(2(1(1(x1))))))))))))))))	→	5(5(1(4(1(4(1(2(5(2(0(5(2(3(1(x1)))))))))))))))	(28)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)
1(3(0(4(0(4(2(3(0(2(3(5(0(5(3(5(3(x1)))))))))))))))))	→	1(0(3(0(1(4(5(1(5(3(5(2(3(2(0(1(3(x1)))))))))))))))))	(30)
3(5(1(3(0(0(3(3(3(3(1(0(3(0(5(2(0(x1)))))))))))))))))	→	3(5(3(4(3(5(0(5(1(2(0(4(1(5(3(2(0(x1)))))))))))))))))	(31)
5(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))	→	5(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))	(32)
0(2(0(4(4(4(3(4(0(3(5(1(5(5(3(1(4(3(x1))))))))))))))))))	→	5(2(2(1(1(2(4(2(3(5(0(0(3(1(1(4(0(x1)))))))))))))))))	(33)
0(4(4(1(2(0(2(2(0(3(3(5(2(2(3(2(3(1(x1))))))))))))))))))	→	4(4(0(4(0(0(0(1(3(0(3(3(4(4(4(0(2(4(x1))))))))))))))))))	(34)
3(0(4(4(4(1(1(2(5(2(1(1(4(3(4(4(4(2(x1))))))))))))))))))	→	3(3(4(5(3(3(5(4(3(3(5(2(3(3(5(3(x1))))))))))))))))	(35)
0(0(4(1(2(3(5(0(4(0(5(2(4(3(2(3(1(5(1(x1)))))))))))))))))))	→	4(5(3(2(0(2(3(4(5(5(1(1(5(1(5(5(1(4(x1))))))))))))))))))	(36)
0(4(2(5(2(1(0(1(0(1(3(3(5(1(1(2(5(0(2(x1)))))))))))))))))))	→	2(4(1(3(3(5(0(2(0(3(3(5(3(0(1(4(5(0(2(x1)))))))))))))))))))	(37)
1(0(0(4(4(4(1(3(2(4(5(1(2(5(3(2(3(2(2(5(2(x1)))))))))))))))))))))	→	1(4(5(1(3(5(4(2(4(4(4(5(0(2(2(1(1(1(1(3(4(x1)))))))))))))))))))))	(38)
3(4(4(4(2(1(5(3(1(4(1(5(3(5(3(3(0(2(1(3(1(x1)))))))))))))))))))))	→	3(4(3(0(3(3(4(4(3(4(0(1(4(0(1(5(1(5(4(2(3(x1)))))))))))))))))))))	(39)
5(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))	→	5(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))	(40)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals

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

all of the following rules can be deleted.

2(1(4(3(1(x1)))))	→	4(1(5(3(x1))))	(2)
0(1(1(3(2(3(1(3(2(x1)))))))))	→	0(0(5(3(3(1(1(0(x1))))))))	(7)
0(1(1(1(2(2(4(1(1(1(x1))))))))))	→	2(1(2(4(0(2(2(5(0(x1)))))))))	(10)
2(2(2(1(4(1(1(3(0(2(x1))))))))))	→	0(2(0(4(2(3(5(4(0(x1)))))))))	(11)
2(2(4(2(1(2(3(2(1(4(x1))))))))))	→	3(3(3(0(4(5(0(2(4(x1)))))))))	(12)
3(2(1(1(4(2(4(5(2(5(3(3(x1))))))))))))	→	5(1(3(1(2(4(1(2(2(1(0(x1)))))))))))	(16)
5(1(3(3(0(3(5(2(1(4(1(5(x1))))))))))))	→	3(5(5(5(5(0(1(5(4(2(5(x1)))))))))))	(17)
1(2(5(2(4(4(0(3(4(0(3(3(2(x1)))))))))))))	→	1(2(4(2(5(3(5(2(1(3(2(0(x1))))))))))))	(20)
5(5(0(2(4(4(2(4(2(2(0(3(4(x1)))))))))))))	→	5(3(5(2(0(0(4(4(5(5(5(2(x1))))))))))))	(21)
2(2(4(0(2(2(4(3(5(0(2(3(3(5(3(x1)))))))))))))))	→	2(1(5(1(3(5(5(5(5(0(0(2(5(x1)))))))))))))	(23)
5(0(2(2(0(2(4(3(2(2(4(5(0(2(3(x1)))))))))))))))	→	5(4(1(1(5(4(5(0(2(3(0(3(3(3(x1))))))))))))))	(24)
3(3(3(4(0(3(1(4(4(4(4(1(3(3(1(4(x1))))))))))))))))	→	5(5(2(4(0(0(3(0(4(5(0(2(2(2(x1))))))))))))))	(27)
3(3(3(5(3(1(1(2(2(3(4(2(0(2(1(1(x1))))))))))))))))	→	5(5(1(4(1(4(1(2(5(2(0(5(2(3(1(x1)))))))))))))))	(28)
0(2(0(4(4(4(3(4(0(3(5(1(5(5(3(1(4(3(x1))))))))))))))))))	→	5(2(2(1(1(2(4(2(3(5(0(0(3(1(1(4(0(x1)))))))))))))))))	(33)
3(0(4(4(4(1(1(2(5(2(1(1(4(3(4(4(4(2(x1))))))))))))))))))	→	3(3(4(5(3(3(5(4(3(3(5(2(3(3(5(3(x1))))))))))))))))	(35)
0(0(4(1(2(3(5(0(4(0(5(2(4(3(2(3(1(5(1(x1)))))))))))))))))))	→	4(5(3(2(0(2(3(4(5(5(1(1(5(1(5(5(1(4(x1))))))))))))))))))	(36)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 301 ruless (increase limit for explicit display).

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pairs

There are 275 ruless (increase limit for explicit display).

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 6 components.

The 1^st component contains the pair

1^#(3(5(3(2(3(4(1(5(1(4(1(0(3(x1))))))))))))))	→	1^#(1(1(0(0(2(0(3(4(2(1(3(0(3(x1))))))))))))))	(140)
1^#(2(3(5(3(0(x1))))))	→	1^#(1(4(3(5(1(x1))))))	(45)
1^#(3(0(4(0(4(2(3(0(2(3(5(0(5(3(5(3(x1)))))))))))))))))	→	1^#(0(3(0(1(4(5(1(5(3(5(2(3(2(0(1(3(x1)))))))))))))))))	(197)
1^#(0(0(4(4(4(1(3(2(4(5(1(2(5(3(2(3(2(2(5(2(x1)))))))))))))))))))))	→	1^#(4(5(1(3(5(4(2(4(4(4(5(0(2(2(1(1(1(1(3(4(x1)))))))))))))))))))))	(279)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

0(1(3(1(5(2(2(4(4(5(5(2(4(x1)))))))))))))	→	0(3(4(0(0(4(0(2(1(3(4(3(4(x1)))))))))))))	(19)
0(1(2(2(3(x1)))))	→	0(4(1(4(3(x1)))))	(1)
0(0(1(2(2(0(4(2(x1))))))))	→	2(4(0(0(4(4(3(0(x1))))))))	(6)
0(1(4(2(0(5(1(5(2(4(0(0(3(4(4(2(x1))))))))))))))))	→	4(1(5(4(1(0(2(0(3(4(4(1(1(3(4(2(x1))))))))))))))))	(25)
0(4(4(1(2(0(2(2(0(3(3(5(2(2(3(2(3(1(x1))))))))))))))))))	→	4(4(0(4(0(0(0(1(3(0(3(3(4(4(4(0(2(4(x1))))))))))))))))))	(34)
0(4(2(5(2(1(0(1(0(1(3(3(5(1(1(2(5(0(2(x1)))))))))))))))))))	→	2(4(1(3(3(5(0(2(0(3(3(5(3(0(1(4(5(0(2(x1)))))))))))))))))))	(37)
1(3(5(3(2(3(4(1(5(1(4(1(0(3(x1))))))))))))))	→	1(1(1(0(0(2(0(3(4(2(1(3(0(3(x1))))))))))))))	(22)
1(2(3(5(3(0(x1))))))	→	1(1(4(3(5(1(x1))))))	(3)
1(3(0(4(0(4(2(3(0(2(3(5(0(5(3(5(3(x1)))))))))))))))))	→	1(0(3(0(1(4(5(1(5(3(5(2(3(2(0(1(3(x1)))))))))))))))))	(30)
1(0(0(4(4(4(1(3(2(4(5(1(2(5(3(2(3(2(2(5(2(x1)))))))))))))))))))))	→	1(4(5(1(3(5(4(2(4(4(4(5(0(2(2(1(1(1(1(3(4(x1)))))))))))))))))))))	(38)
1(2(5(5(5(2(5(3(5(x1)))))))))	→	4(5(3(2(5(1(3(2(5(x1)))))))))	(8)
1(3(1(5(0(4(2(2(0(5(5(x1)))))))))))	→	4(4(1(2(2(5(2(2(2(0(5(x1)))))))))))	(13)
2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)
5(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))	→	5(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))	(32)
5(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))	→	5(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))	(18)
5(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))	→	5(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))	(40)

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

1^#(0(0(4(4(4(1(3(2(4(5(1(2(5(3(2(3(2(2(5(2(x1)))))))))))))))))))))

→

1^#(4(5(1(3(5(4(2(4(4(4(5(0(2(2(1(1(1(1(3(4(x1)))))))))))))))))))))

(279)

could be deleted.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

0(1(3(1(5(2(2(4(4(5(5(2(4(x1)))))))))))))	→	0(3(4(0(0(4(0(2(1(3(4(3(4(x1)))))))))))))	(19)
0(1(2(2(3(x1)))))	→	0(4(1(4(3(x1)))))	(1)
0(0(1(2(2(0(4(2(x1))))))))	→	2(4(0(0(4(4(3(0(x1))))))))	(6)
0(1(4(2(0(5(1(5(2(4(0(0(3(4(4(2(x1))))))))))))))))	→	4(1(5(4(1(0(2(0(3(4(4(1(1(3(4(2(x1))))))))))))))))	(25)
0(4(4(1(2(0(2(2(0(3(3(5(2(2(3(2(3(1(x1))))))))))))))))))	→	4(4(0(4(0(0(0(1(3(0(3(3(4(4(4(0(2(4(x1))))))))))))))))))	(34)
0(4(2(5(2(1(0(1(0(1(3(3(5(1(1(2(5(0(2(x1)))))))))))))))))))	→	2(4(1(3(3(5(0(2(0(3(3(5(3(0(1(4(5(0(2(x1)))))))))))))))))))	(37)
1(3(5(3(2(3(4(1(5(1(4(1(0(3(x1))))))))))))))	→	1(1(1(0(0(2(0(3(4(2(1(3(0(3(x1))))))))))))))	(22)
1(2(3(5(3(0(x1))))))	→	1(1(4(3(5(1(x1))))))	(3)
1(3(0(4(0(4(2(3(0(2(3(5(0(5(3(5(3(x1)))))))))))))))))	→	1(0(3(0(1(4(5(1(5(3(5(2(3(2(0(1(3(x1)))))))))))))))))	(30)
1(0(0(4(4(4(1(3(2(4(5(1(2(5(3(2(3(2(2(5(2(x1)))))))))))))))))))))	→	1(4(5(1(3(5(4(2(4(4(4(5(0(2(2(1(1(1(1(3(4(x1)))))))))))))))))))))	(38)
1(2(5(5(5(2(5(3(5(x1)))))))))	→	4(5(3(2(5(1(3(2(5(x1)))))))))	(8)
1(3(1(5(0(4(2(2(0(5(5(x1)))))))))))	→	4(4(1(2(2(5(2(2(2(0(5(x1)))))))))))	(13)
2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)

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

1^#(3(5(3(2(3(4(1(5(1(4(1(0(3(x1))))))))))))))	→	1^#(1(1(0(0(2(0(3(4(2(1(3(0(3(x1))))))))))))))	(140)
1^#(2(3(5(3(0(x1))))))	→	1^#(1(4(3(5(1(x1))))))	(45)
1^#(3(0(4(0(4(2(3(0(2(3(5(0(5(3(5(3(x1)))))))))))))))))	→	1^#(0(3(0(1(4(5(1(5(3(5(2(3(2(0(1(3(x1)))))))))))))))))	(197)

could be deleted.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

4^#(1(5(1(2(4(1(4(1(x1)))))))))	→	4^#(2(3(1(3(0(1(3(1(x1)))))))))	(78)
4^#(3(5(1(5(1(x1))))))	→	4^#(0(2(4(5(4(x1))))))	(56)
4^#(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4^#(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(105)
4^#(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4^#(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(181)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

0(1(3(1(5(2(2(4(4(5(5(2(4(x1)))))))))))))	→	0(3(4(0(0(4(0(2(1(3(4(3(4(x1)))))))))))))	(19)
0(1(2(2(3(x1)))))	→	0(4(1(4(3(x1)))))	(1)
0(0(1(2(2(0(4(2(x1))))))))	→	2(4(0(0(4(4(3(0(x1))))))))	(6)
0(1(4(2(0(5(1(5(2(4(0(0(3(4(4(2(x1))))))))))))))))	→	4(1(5(4(1(0(2(0(3(4(4(1(1(3(4(2(x1))))))))))))))))	(25)
0(4(4(1(2(0(2(2(0(3(3(5(2(2(3(2(3(1(x1))))))))))))))))))	→	4(4(0(4(0(0(0(1(3(0(3(3(4(4(4(0(2(4(x1))))))))))))))))))	(34)
0(4(2(5(2(1(0(1(0(1(3(3(5(1(1(2(5(0(2(x1)))))))))))))))))))	→	2(4(1(3(3(5(0(2(0(3(3(5(3(0(1(4(5(0(2(x1)))))))))))))))))))	(37)
2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)

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

4^#(3(5(1(5(1(x1))))))	→	4^#(0(2(4(5(4(x1))))))	(56)
4^#(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4^#(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(181)

could be deleted.

1.1.1.1.2.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)

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

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

→

4^#(4(4(3(3(2(0(2(4(5(3(x1)))))))))))

(105)

could be deleted.

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)

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

4^#(1(5(1(2(4(1(4(1(x1)))))))))

→

4^#(2(3(1(3(0(1(3(1(x1)))))))))

(78)

could be deleted.

1.1.1.1.2.1.1.1 P is empty

There are no pairs anymore.

The 3^rd component contains the pair

0^#(1(3(1(5(2(2(4(4(5(5(2(4(x1)))))))))))))	→	0^#(3(4(0(0(4(0(2(1(3(4(3(4(x1)))))))))))))	(128)
0^#(1(2(2(3(x1)))))	→	0^#(4(1(4(3(x1)))))	(41)

1.1.1.1.3 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)
3(4(4(4(2(1(5(3(1(4(1(5(3(5(3(3(0(2(1(3(1(x1)))))))))))))))))))))	→	3(4(3(0(3(3(4(4(3(4(0(1(4(0(1(5(1(5(4(2(3(x1)))))))))))))))))))))	(39)
3(5(1(3(0(0(3(3(3(3(1(0(3(0(5(2(0(x1)))))))))))))))))	→	3(5(3(4(3(5(0(5(1(2(0(4(1(5(3(2(0(x1)))))))))))))))))	(31)

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

0^#(1(2(2(3(x1)))))

→

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

(41)

could be deleted.

1.1.1.1.3.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

3(4(4(4(2(1(5(3(1(4(1(5(3(5(3(3(0(2(1(3(1(x1)))))))))))))))))))))	→	3(4(3(0(3(3(4(4(3(4(0(1(4(0(1(5(1(5(4(2(3(x1)))))))))))))))))))))	(39)
3(5(1(3(0(0(3(3(3(3(1(0(3(0(5(2(0(x1)))))))))))))))))	→	3(5(3(4(3(5(0(5(1(2(0(4(1(5(3(2(0(x1)))))))))))))))))	(31)

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

0^#(1(3(1(5(2(2(4(4(5(5(2(4(x1)))))))))))))

→

0^#(3(4(0(0(4(0(2(1(3(4(3(4(x1)))))))))))))

(128)

could be deleted.

1.1.1.1.3.1.1 P is empty

There are no pairs anymore.

The 4^th component contains the pair

5^#(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))	→	5^#(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))	(228)
5^#(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))	→	5^#(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))	(116)
5^#(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))	→	5^#(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))	(321)

1.1.1.1.4 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)
5(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))	→	5(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))	(32)
5(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))	→	5(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))	(18)
5(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))	→	5(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))	(40)

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

5^#(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))

→

5^#(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))

(116)

could be deleted.

1.1.1.1.4.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pair

5^#(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))

→

5^#(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))

(321)

could be deleted.

1.1.1.1.4.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pair

5^#(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))

→

5^#(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))

(228)

could be deleted.

1.1.1.1.4.1.1.1 P is empty

There are no pairs anymore.

The 5^th component contains the pair

2^#(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2^#(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(166)
2^#(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2^#(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(96)

1.1.1.1.5 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)
5(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))	→	5(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))	(32)
5(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))	→	5(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))	(18)
5(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))	→	5(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))	(40)
1(3(5(3(2(3(4(1(5(1(4(1(0(3(x1))))))))))))))	→	1(1(1(0(0(2(0(3(4(2(1(3(0(3(x1))))))))))))))	(22)
1(2(3(5(3(0(x1))))))	→	1(1(4(3(5(1(x1))))))	(3)
1(3(0(4(0(4(2(3(0(2(3(5(0(5(3(5(3(x1)))))))))))))))))	→	1(0(3(0(1(4(5(1(5(3(5(2(3(2(0(1(3(x1)))))))))))))))))	(30)
1(0(0(4(4(4(1(3(2(4(5(1(2(5(3(2(3(2(2(5(2(x1)))))))))))))))))))))	→	1(4(5(1(3(5(4(2(4(4(4(5(0(2(2(1(1(1(1(3(4(x1)))))))))))))))))))))	(38)
1(2(5(5(5(2(5(3(5(x1)))))))))	→	4(5(3(2(5(1(3(2(5(x1)))))))))	(8)
1(3(1(5(0(4(2(2(0(5(5(x1)))))))))))	→	4(4(1(2(2(5(2(2(2(0(5(x1)))))))))))	(13)

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

2^#(0(1(3(1(2(5(4(3(4(4(x1)))))))))))

→

2^#(1(2(4(4(2(3(4(2(4(4(x1)))))))))))

(96)

could be deleted.

1.1.1.1.5.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

5(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))	→	5(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))	(32)
5(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))	→	5(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))	(18)
5(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))	→	5(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))	(40)

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

2^#(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))

→

2^#(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))

(166)

could be deleted.

1.1.1.1.5.1.1 P is empty

There are no pairs anymore.

The 6^th component contains the pair

3^#(4(4(4(2(1(5(3(1(4(1(5(3(5(3(3(0(2(1(3(1(x1)))))))))))))))))))))	→	3^#(4(3(0(3(3(4(4(3(4(0(1(4(0(1(5(1(5(4(2(3(x1)))))))))))))))))))))	(300)
3^#(5(1(3(0(0(3(3(3(3(1(0(3(0(5(2(0(x1)))))))))))))))))	→	3^#(5(3(4(3(5(0(5(1(2(0(4(1(5(3(2(0(x1)))))))))))))))))	(213)

1.1.1.1.6 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

3(4(4(4(2(1(5(3(1(4(1(5(3(5(3(3(0(2(1(3(1(x1)))))))))))))))))))))	→	3(4(3(0(3(3(4(4(3(4(0(1(4(0(1(5(1(5(4(2(3(x1)))))))))))))))))))))	(39)
3(5(1(3(0(0(3(3(3(3(1(0(3(0(5(2(0(x1)))))))))))))))))	→	3(5(3(4(3(5(0(5(1(2(0(4(1(5(3(2(0(x1)))))))))))))))))	(31)
2(1(3(5(0(0(x1))))))	→	4(4(2(4(5(0(x1))))))	(4)
2(2(5(1(3(4(5(3(1(4(1(2(3(4(2(2(x1))))))))))))))))	→	2(5(2(0(4(2(4(4(2(3(4(0(0(4(3(2(x1))))))))))))))))	(26)
2(0(1(3(1(2(5(4(3(4(4(x1)))))))))))	→	2(1(2(4(4(2(3(4(2(4(4(x1)))))))))))	(14)
4(1(5(1(2(4(1(4(1(x1)))))))))	→	4(2(3(1(3(0(1(3(1(x1)))))))))	(9)
4(3(5(1(5(1(x1))))))	→	4(0(2(4(5(4(x1))))))	(5)
4(1(0(5(3(4(3(5(4(1(1(x1)))))))))))	→	4(4(4(3(3(2(0(2(4(5(3(x1)))))))))))	(15)
4(3(0(3(2(4(5(0(2(5(0(0(4(2(3(1(x1))))))))))))))))	→	4(4(4(2(3(3(4(5(0(5(2(2(1(2(0(3(x1))))))))))))))))	(29)
5(4(5(3(0(3(3(3(1(2(0(0(2(3(3(2(0(x1)))))))))))))))))	→	5(2(0(1(4(2(4(1(2(0(0(4(2(3(1(2(2(x1)))))))))))))))))	(32)
5(3(3(5(3(4(0(5(0(2(4(0(x1))))))))))))	→	5(5(3(2(5(4(4(2(1(1(1(3(x1))))))))))))	(18)
5(2(1(2(2(4(5(2(2(3(3(3(1(0(3(4(1(0(0(3(2(x1)))))))))))))))))))))	→	5(2(3(2(4(0(2(2(3(1(4(3(2(1(2(1(2(5(5(2(4(x1)))))))))))))))))))))	(40)

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

3^#(5(1(3(0(0(3(3(3(3(1(0(3(0(5(2(0(x1)))))))))))))))))

→

3^#(5(3(4(3(5(0(5(1(2(0(4(1(5(3(2(0(x1)))))))))))))))))

(213)

could be deleted.

1.1.1.1.6.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pair

3^#(4(4(4(2(1(5(3(1(4(1(5(3(5(3(3(0(2(1(3(1(x1)))))))))))))))))))))

→

3^#(4(3(0(3(3(4(4(3(4(0(1(4(0(1(5(1(5(4(2(3(x1)))))))))))))))))))))

(300)

could be deleted.

1.1.1.1.6.1.1 P is empty

There are no pairs anymore.