Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212693)

The rewrite relation of the following TRS is considered.

0(1(2(x1)))	→	0(0(2(1(x1))))	(1)
0(1(2(x1)))	→	0(2(1(3(x1))))	(2)
0(1(2(x1)))	→	0(0(2(1(4(4(x1))))))	(3)
0(3(1(x1)))	→	0(1(3(4(0(x1)))))	(4)
0(3(1(x1)))	→	0(1(3(4(4(x1)))))	(5)
0(3(1(x1)))	→	1(3(4(4(4(0(x1))))))	(6)
0(3(2(x1)))	→	0(2(1(3(x1))))	(7)
0(3(2(x1)))	→	0(2(3(4(x1))))	(8)
0(3(2(x1)))	→	0(0(2(4(3(x1)))))	(9)
0(3(2(x1)))	→	0(2(1(4(3(x1)))))	(10)
0(3(2(x1)))	→	0(2(4(3(3(x1)))))	(11)
0(3(2(x1)))	→	0(2(1(3(3(4(x1))))))	(12)
0(3(2(x1)))	→	0(2(3(4(5(5(x1))))))	(13)
0(3(2(x1)))	→	2(4(4(3(4(0(x1))))))	(14)
0(4(1(x1)))	→	0(1(4(4(x1))))	(15)
0(4(1(x1)))	→	0(2(1(4(x1))))	(16)
0(4(2(x1)))	→	0(2(1(4(x1))))	(17)
0(4(2(x1)))	→	0(2(3(4(x1))))	(18)
0(4(2(x1)))	→	0(2(4(3(x1))))	(19)
2(0(1(x1)))	→	5(0(2(1(x1))))	(20)
2(3(1(x1)))	→	1(3(5(2(x1))))	(21)
2(3(1(x1)))	→	0(2(1(3(5(x1)))))	(22)
2(3(1(x1)))	→	1(4(3(5(2(x1)))))	(23)
0(2(0(1(x1))))	→	5(0(0(2(1(x1)))))	(24)
0(3(1(1(x1))))	→	0(1(4(1(3(4(x1))))))	(25)
0(3(2(1(x1))))	→	0(0(3(4(2(1(x1))))))	(26)
0(3(2(2(x1))))	→	1(3(4(0(2(2(x1))))))	(27)
0(4(1(2(x1))))	→	1(4(0(2(5(x1)))))	(28)
0(4(3(2(x1))))	→	2(3(4(4(0(0(x1))))))	(29)
0(5(3(1(x1))))	→	0(1(4(3(5(4(x1))))))	(30)
0(5(3(1(x1))))	→	0(1(5(3(4(0(x1))))))	(31)
0(5(3(2(x1))))	→	0(2(4(5(3(x1)))))	(32)
0(5(3(2(x1))))	→	0(2(5(3(3(x1)))))	(33)
2(0(3(1(x1))))	→	2(0(1(3(5(2(x1))))))	(34)
2(0(4(1(x1))))	→	2(0(1(4(5(x1)))))	(35)
2(5(3(2(x1))))	→	2(5(2(3(3(x1)))))	(36)
2(5(4(2(x1))))	→	0(2(5(2(4(x1)))))	(37)
0(0(3(2(1(x1)))))	→	0(0(1(3(5(2(x1))))))	(38)
0(1(0(3(2(x1)))))	→	0(1(4(3(2(0(x1))))))	(39)
0(1(0(3(2(x1)))))	→	2(3(1(0(0(5(x1))))))	(40)
0(3(2(5(1(x1)))))	→	0(2(5(1(3(3(x1))))))	(41)
0(5(1(1(2(x1)))))	→	0(2(4(1(1(5(x1))))))	(42)
0(5(1(2(2(x1)))))	→	0(2(5(2(1(2(x1))))))	(43)
0(5(3(2(1(x1)))))	→	0(1(3(4(2(5(x1))))))	(44)
0(5(5(3(2(x1)))))	→	0(2(5(1(3(5(x1))))))	(45)
2(0(3(1(1(x1)))))	→	2(1(0(1(3(4(x1))))))	(46)
2(2(0(3(1(x1)))))	→	1(3(0(2(5(2(x1))))))	(47)
2(2(0(5(1(x1)))))	→	2(0(2(1(5(1(x1))))))	(48)
2(5(5(4(1(x1)))))	→	5(5(2(1(3(4(x1))))))	(49)

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.

There are 102 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(5(3(1(x1))))	→	0^#(x1)	(108)
0^#(1(0(3(2(x1)))))	→	2^#(3(1(0(0(5(x1))))))	(106)
0^#(1(0(3(2(x1)))))	→	0^#(0(5(x1)))	(99)
0^#(4(1(2(x1))))	→	0^#(2(5(x1)))	(96)
0^#(4(3(2(x1))))	→	0^#(x1)	(143)
0^#(3(2(2(x1))))	→	0^#(2(2(x1)))	(87)
0^#(3(1(x1)))	→	0^#(x1)	(53)
0^#(3(2(x1)))	→	0^#(x1)	(81)
0^#(1(0(3(2(x1)))))	→	0^#(5(x1))	(130)
0^#(1(0(3(2(x1)))))	→	2^#(0(x1))	(128)
0^#(1(0(3(2(x1)))))	→	0^#(x1)	(73)
2^#(3(1(x1)))	→	2^#(x1)	(52)
2^#(0(3(1(x1))))	→	2^#(x1)	(70)
2^#(2(0(3(1(x1)))))	→	2^#(x1)	(124)
0^#(0(3(2(1(x1)))))	→	2^#(x1)	(123)
0^#(4(3(2(x1))))	→	0^#(0(x1))	(112)
0^#(3(1(x1)))	→	0^#(x1)	(53)
2^#(3(1(x1)))	→	2^#(x1)	(52)
2^#(2(0(3(1(x1)))))	→	0^#(2(5(2(x1))))	(110)

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)]

0	1
0	0

· x₁ +

68983

[1(x₁)]

0	0
1	1

· x₁ +

34492

[4(x₁)]

0	0
0	1

· x₁ +

[5(x₁)]

0	0
0	1

· x₁ +

[3(x₁)]

0	0
0	1

· x₁ +

[2^#(x₁)]

1	1
0	0

· x₁ +

[0(x₁)]

0	0
0	1

· x₁ +

34492

[2(x₁)]

x₁ +

34492

together with the usable rules

0(4(2(x1)))	→	0(2(3(4(x1))))	(18)
0(3(1(x1)))	→	0(1(3(4(0(x1)))))	(4)
0(4(1(x1)))	→	0(1(4(4(x1))))	(15)
0(3(2(x1)))	→	0(2(3(4(x1))))	(8)
0(1(2(x1)))	→	0(0(2(1(x1))))	(1)
0(1(2(x1)))	→	0(0(2(1(4(4(x1))))))	(3)
0(4(1(x1)))	→	0(2(1(4(x1))))	(16)
2(3(1(x1)))	→	1(3(5(2(x1))))	(21)
2(5(3(2(x1))))	→	2(5(2(3(3(x1)))))	(36)
0(3(2(1(x1))))	→	0(0(3(4(2(1(x1))))))	(26)
0(4(2(x1)))	→	0(2(4(3(x1))))	(19)
0(5(3(2(x1))))	→	0(2(4(5(3(x1)))))	(32)
0(4(2(x1)))	→	0(2(1(4(x1))))	(17)
0(3(2(2(x1))))	→	1(3(4(0(2(2(x1))))))	(27)
2(0(3(1(x1))))	→	2(0(1(3(5(2(x1))))))	(34)
2(3(1(x1)))	→	0(2(1(3(5(x1)))))	(22)
0(4(1(2(x1))))	→	1(4(0(2(5(x1)))))	(28)
0(5(3(2(1(x1)))))	→	0(1(3(4(2(5(x1))))))	(44)
0(3(1(x1)))	→	0(1(3(4(4(x1)))))	(5)
0(5(3(2(x1))))	→	0(2(5(3(3(x1)))))	(33)
0(3(2(x1)))	→	0(2(1(4(3(x1)))))	(10)
0(1(0(3(2(x1)))))	→	0(1(4(3(2(0(x1))))))	(39)
0(3(2(x1)))	→	0(2(1(3(x1))))	(7)
2(0(1(x1)))	→	5(0(2(1(x1))))	(20)
0(3(1(1(x1))))	→	0(1(4(1(3(4(x1))))))	(25)
2(5(5(4(1(x1)))))	→	5(5(2(1(3(4(x1))))))	(49)
0(5(3(1(x1))))	→	0(1(4(3(5(4(x1))))))	(30)
0(3(2(x1)))	→	2(4(4(3(4(0(x1))))))	(14)
0(5(3(1(x1))))	→	0(1(5(3(4(0(x1))))))	(31)
0(3(2(x1)))	→	0(2(1(3(3(4(x1))))))	(12)
0(5(5(3(2(x1)))))	→	0(2(5(1(3(5(x1))))))	(45)
2(3(1(x1)))	→	1(4(3(5(2(x1)))))	(23)
0(2(0(1(x1))))	→	5(0(0(2(1(x1)))))	(24)
0(3(2(x1)))	→	0(2(4(3(3(x1)))))	(11)
0(3(2(x1)))	→	0(0(2(4(3(x1)))))	(9)
0(3(2(x1)))	→	0(2(3(4(5(5(x1))))))	(13)
0(1(0(3(2(x1)))))	→	2(3(1(0(0(5(x1))))))	(40)
0(3(1(x1)))	→	1(3(4(4(4(0(x1))))))	(6)
0(0(3(2(1(x1)))))	→	0(0(1(3(5(2(x1))))))	(38)
2(2(0(5(1(x1)))))	→	2(0(2(1(5(1(x1))))))	(48)
2(2(0(3(1(x1)))))	→	1(3(0(2(5(2(x1))))))	(47)
2(5(4(2(x1))))	→	0(2(5(2(4(x1)))))	(37)
0(3(2(5(1(x1)))))	→	0(2(5(1(3(3(x1))))))	(41)
0(5(1(1(2(x1)))))	→	0(2(4(1(1(5(x1))))))	(42)
2(0(3(1(1(x1)))))	→	2(1(0(1(3(4(x1))))))	(46)
2(0(4(1(x1))))	→	2(0(1(4(5(x1)))))	(35)
0(4(3(2(x1))))	→	2(3(4(4(0(0(x1))))))	(29)
0(5(1(2(2(x1)))))	→	0(2(5(2(1(2(x1))))))	(43)
0(1(2(x1)))	→	0(2(1(3(x1))))	(2)

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

0^#(1(0(3(2(x1)))))	→	2^#(3(1(0(0(5(x1))))))	(106)
0^#(1(0(3(2(x1)))))	→	0^#(0(5(x1)))	(99)
0^#(4(1(2(x1))))	→	0^#(2(5(x1)))	(96)
0^#(1(0(3(2(x1)))))	→	0^#(5(x1))	(130)
0^#(1(0(3(2(x1)))))	→	2^#(0(x1))	(128)
0^#(1(0(3(2(x1)))))	→	0^#(x1)	(73)
2^#(0(3(1(x1))))	→	2^#(x1)	(70)
2^#(2(0(3(1(x1)))))	→	2^#(x1)	(124)
0^#(0(3(2(1(x1)))))	→	2^#(x1)	(123)
2^#(2(0(3(1(x1)))))	→	0^#(2(5(2(x1))))	(110)

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

2^#(3(1(x1)))	→	2^#(x1)	(52)
2^#(3(1(x1)))	→	2^#(x1)	(52)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

68983

[1(x₁)]

0	0
1	0

· x₁ +

57280

[4(x₁)]

[5(x₁)]

2283

[3(x₁)]

1	1
1	0

· x₁ +

13170

2282

[2^#(x₁)]

1	0
0	0

· x₁ +

[0(x₁)]

3132

72734

[2(x₁)]

0	1
0	1

· x₁ +

849

70451

together with the usable rules

0(4(2(x1)))	→	0(2(3(4(x1))))	(18)
0(3(1(x1)))	→	0(1(3(4(0(x1)))))	(4)
0(4(1(x1)))	→	0(1(4(4(x1))))	(15)
0(3(2(x1)))	→	0(2(3(4(x1))))	(8)
0(1(2(x1)))	→	0(0(2(1(x1))))	(1)
0(1(2(x1)))	→	0(0(2(1(4(4(x1))))))	(3)
0(4(1(x1)))	→	0(2(1(4(x1))))	(16)
2(3(1(x1)))	→	1(3(5(2(x1))))	(21)
2(5(3(2(x1))))	→	2(5(2(3(3(x1)))))	(36)
0(3(2(1(x1))))	→	0(0(3(4(2(1(x1))))))	(26)
0(4(2(x1)))	→	0(2(4(3(x1))))	(19)
0(5(3(2(x1))))	→	0(2(4(5(3(x1)))))	(32)
0(4(2(x1)))	→	0(2(1(4(x1))))	(17)
0(3(2(2(x1))))	→	1(3(4(0(2(2(x1))))))	(27)
2(0(3(1(x1))))	→	2(0(1(3(5(2(x1))))))	(34)
2(3(1(x1)))	→	0(2(1(3(5(x1)))))	(22)
0(4(1(2(x1))))	→	1(4(0(2(5(x1)))))	(28)
0(5(3(2(1(x1)))))	→	0(1(3(4(2(5(x1))))))	(44)
0(3(1(x1)))	→	0(1(3(4(4(x1)))))	(5)
0(5(3(2(x1))))	→	0(2(5(3(3(x1)))))	(33)
0(3(2(x1)))	→	0(2(1(4(3(x1)))))	(10)
0(1(0(3(2(x1)))))	→	0(1(4(3(2(0(x1))))))	(39)
0(3(2(x1)))	→	0(2(1(3(x1))))	(7)
2(0(1(x1)))	→	5(0(2(1(x1))))	(20)
0(3(1(1(x1))))	→	0(1(4(1(3(4(x1))))))	(25)
2(5(5(4(1(x1)))))	→	5(5(2(1(3(4(x1))))))	(49)
0(5(3(1(x1))))	→	0(1(4(3(5(4(x1))))))	(30)
0(3(2(x1)))	→	2(4(4(3(4(0(x1))))))	(14)
0(5(3(1(x1))))	→	0(1(5(3(4(0(x1))))))	(31)
0(3(2(x1)))	→	0(2(1(3(3(4(x1))))))	(12)
0(5(5(3(2(x1)))))	→	0(2(5(1(3(5(x1))))))	(45)
2(3(1(x1)))	→	1(4(3(5(2(x1)))))	(23)
0(2(0(1(x1))))	→	5(0(0(2(1(x1)))))	(24)
0(3(2(x1)))	→	0(2(4(3(3(x1)))))	(11)
0(3(2(x1)))	→	0(0(2(4(3(x1)))))	(9)
0(3(2(x1)))	→	0(2(3(4(5(5(x1))))))	(13)
0(1(0(3(2(x1)))))	→	2(3(1(0(0(5(x1))))))	(40)
0(3(1(x1)))	→	1(3(4(4(4(0(x1))))))	(6)
0(0(3(2(1(x1)))))	→	0(0(1(3(5(2(x1))))))	(38)
2(2(0(5(1(x1)))))	→	2(0(2(1(5(1(x1))))))	(48)
2(2(0(3(1(x1)))))	→	1(3(0(2(5(2(x1))))))	(47)
2(5(4(2(x1))))	→	0(2(5(2(4(x1)))))	(37)
0(3(2(5(1(x1)))))	→	0(2(5(1(3(3(x1))))))	(41)
0(5(1(1(2(x1)))))	→	0(2(4(1(1(5(x1))))))	(42)
2(0(3(1(1(x1)))))	→	2(1(0(1(3(4(x1))))))	(46)
2(0(4(1(x1))))	→	2(0(1(4(5(x1)))))	(35)
0(4(3(2(x1))))	→	2(3(4(4(0(0(x1))))))	(29)
0(5(1(2(2(x1)))))	→	0(2(5(2(1(2(x1))))))	(43)
0(1(2(x1)))	→	0(2(1(3(x1))))	(2)

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

2^#(3(1(x1)))	→	2^#(x1)	(52)
2^#(3(1(x1)))	→	2^#(x1)	(52)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

0^#(3(1(x1)))	→	0^#(x1)	(53)
0^#(3(2(2(x1))))	→	0^#(2(2(x1)))	(87)
0^#(3(2(x1)))	→	0^#(x1)	(81)
0^#(5(3(1(x1))))	→	0^#(x1)	(108)
0^#(3(1(x1)))	→	0^#(x1)	(53)
0^#(4(3(2(x1))))	→	0^#(x1)	(143)
0^#(4(3(2(x1))))	→	0^#(0(x1))	(112)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[0^#(x₁)]

1	0
0	0

· x₁ +

68983

[1(x₁)]

0	0
1	0

· x₁ +

58459

[4(x₁)]

1	0
0	0

· x₁ +

18960

39499

[5(x₁)]

1	0
0	0

· x₁ +

122521

[3(x₁)]

1	1
0	1

· x₁ +

5603

18960

[2^#(x₁)]

[0(x₁)]

205543

[2(x₁)]

0	0
1	1

· x₁ +

64062

together with the usable rules

0(4(2(x1)))	→	0(2(3(4(x1))))	(18)
0(3(1(x1)))	→	0(1(3(4(0(x1)))))	(4)
0(4(1(x1)))	→	0(1(4(4(x1))))	(15)
0(3(2(x1)))	→	0(2(3(4(x1))))	(8)
0(1(2(x1)))	→	0(0(2(1(x1))))	(1)
0(1(2(x1)))	→	0(0(2(1(4(4(x1))))))	(3)
0(4(1(x1)))	→	0(2(1(4(x1))))	(16)
2(3(1(x1)))	→	1(3(5(2(x1))))	(21)
2(5(3(2(x1))))	→	2(5(2(3(3(x1)))))	(36)
0(3(2(1(x1))))	→	0(0(3(4(2(1(x1))))))	(26)
0(4(2(x1)))	→	0(2(4(3(x1))))	(19)
0(5(3(2(x1))))	→	0(2(4(5(3(x1)))))	(32)
0(4(2(x1)))	→	0(2(1(4(x1))))	(17)
0(3(2(2(x1))))	→	1(3(4(0(2(2(x1))))))	(27)
2(0(3(1(x1))))	→	2(0(1(3(5(2(x1))))))	(34)
2(3(1(x1)))	→	0(2(1(3(5(x1)))))	(22)
0(4(1(2(x1))))	→	1(4(0(2(5(x1)))))	(28)
0(5(3(2(1(x1)))))	→	0(1(3(4(2(5(x1))))))	(44)
0(3(1(x1)))	→	0(1(3(4(4(x1)))))	(5)
0(5(3(2(x1))))	→	0(2(5(3(3(x1)))))	(33)
0(3(2(x1)))	→	0(2(1(4(3(x1)))))	(10)
0(1(0(3(2(x1)))))	→	0(1(4(3(2(0(x1))))))	(39)
0(3(2(x1)))	→	0(2(1(3(x1))))	(7)
2(0(1(x1)))	→	5(0(2(1(x1))))	(20)
0(3(1(1(x1))))	→	0(1(4(1(3(4(x1))))))	(25)
2(5(5(4(1(x1)))))	→	5(5(2(1(3(4(x1))))))	(49)
0(5(3(1(x1))))	→	0(1(4(3(5(4(x1))))))	(30)
0(3(2(x1)))	→	2(4(4(3(4(0(x1))))))	(14)
0(5(3(1(x1))))	→	0(1(5(3(4(0(x1))))))	(31)
0(3(2(x1)))	→	0(2(1(3(3(4(x1))))))	(12)
0(5(5(3(2(x1)))))	→	0(2(5(1(3(5(x1))))))	(45)
2(3(1(x1)))	→	1(4(3(5(2(x1)))))	(23)
0(2(0(1(x1))))	→	5(0(0(2(1(x1)))))	(24)
0(3(2(x1)))	→	0(2(4(3(3(x1)))))	(11)
0(3(2(x1)))	→	0(0(2(4(3(x1)))))	(9)
0(3(2(x1)))	→	0(2(3(4(5(5(x1))))))	(13)
0(1(0(3(2(x1)))))	→	2(3(1(0(0(5(x1))))))	(40)
0(3(1(x1)))	→	1(3(4(4(4(0(x1))))))	(6)
0(0(3(2(1(x1)))))	→	0(0(1(3(5(2(x1))))))	(38)
2(2(0(5(1(x1)))))	→	2(0(2(1(5(1(x1))))))	(48)
2(2(0(3(1(x1)))))	→	1(3(0(2(5(2(x1))))))	(47)
2(5(4(2(x1))))	→	0(2(5(2(4(x1)))))	(37)
0(3(2(5(1(x1)))))	→	0(2(5(1(3(3(x1))))))	(41)
0(5(1(1(2(x1)))))	→	0(2(4(1(1(5(x1))))))	(42)
2(0(3(1(1(x1)))))	→	2(1(0(1(3(4(x1))))))	(46)
2(0(4(1(x1))))	→	2(0(1(4(5(x1)))))	(35)
0(4(3(2(x1))))	→	2(3(4(4(0(0(x1))))))	(29)
0(5(1(2(2(x1)))))	→	0(2(5(2(1(2(x1))))))	(43)
0(1(2(x1)))	→	0(2(1(3(x1))))	(2)

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

0^#(3(1(x1)))	→	0^#(x1)	(53)
0^#(3(2(2(x1))))	→	0^#(2(2(x1)))	(87)
0^#(3(2(x1)))	→	0^#(x1)	(81)
0^#(5(3(1(x1))))	→	0^#(x1)	(108)
0^#(3(1(x1)))	→	0^#(x1)	(53)
0^#(4(3(2(x1))))	→	0^#(x1)	(143)
0^#(4(3(2(x1))))	→	0^#(0(x1))	(112)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.