Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213719)

The rewrite relation of the following TRS is considered.

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

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 166 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

5^#(0(4(2(3(x1)))))	→	4^#(1(5(3(x1))))	(205)
0^#(0(4(2(3(x1)))))	→	4^#(x1)	(138)
0^#(1(0(2(3(x1)))))	→	4^#(1(0(0(x1))))	(203)
0^#(0(5(2(3(x1)))))	→	4^#(0(x1))	(137)
5^#(3(1(2(3(x1)))))	→	1^#(5(3(x1)))	(133)
0^#(4(5(5(3(x1)))))	→	5^#(x1)	(131)
5^#(0(4(2(3(x1)))))	→	5^#(x1)	(199)
0^#(1(0(5(3(x1)))))	→	4^#(x1)	(129)
0^#(1(0(2(3(x1)))))	→	0^#(0(x1))	(128)
5^#(0(1(2(3(x1)))))	→	1^#(x1)	(126)
0^#(1(2(3(3(x1)))))	→	1^#(x1)	(124)
5^#(3(1(2(3(x1)))))	→	5^#(3(x1))	(122)
1^#(0(0(5(3(x1)))))	→	0^#(x1)	(119)
0^#(5(2(1(3(x1)))))	→	5^#(x1)	(193)
1^#(0(1(2(3(x1)))))	→	1^#(x1)	(118)
0^#(1(2(3(3(x1)))))	→	4^#(1(x1))	(116)
0^#(4(5(5(3(x1)))))	→	4^#(1(5(x1)))	(115)
0^#(1(0(2(3(x1)))))	→	1^#(0(0(x1)))	(114)
0^#(0(1(2(3(x1)))))	→	0^#(x1)	(156)
0^#(1(2(5(3(x1)))))	→	1^#(0(x1))	(110)
0^#(5(4(2(3(x1)))))	→	1^#(x1)	(106)
0^#(0(5(2(3(x1)))))	→	0^#(x1)	(182)
0^#(1(2(1(3(x1)))))	→	1^#(x1)	(105)
1^#(0(0(5(3(x1)))))	→	5^#(0(x1))	(181)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
5^#(0(0(5(3(x1)))))	→	5^#(5(0(x1)))	(178)
0^#(1(2(5(3(x1)))))	→	0^#(x1)	(176)
0^#(1(5(3(x1))))	→	5^#(x1)	(99)
5^#(0(4(2(3(x1)))))	→	1^#(5(3(x1)))	(172)
0^#(1(2(3(x1))))	→	0^#(x1)	(95)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
0^#(0(4(2(3(x1)))))	→	4^#(4(x1))	(89)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
4^#(0(1(2(3(x1)))))	→	1^#(0(0(x1)))	(165)
0^#(1(2(3(x1))))	→	1^#(1(x1))	(86)
5^#(0(0(5(3(x1)))))	→	5^#(0(x1))	(164)
0^#(0(1(2(3(x1)))))	→	1^#(0(x1))	(82)
4^#(0(1(2(3(x1)))))	→	0^#(0(x1))	(80)
0^#(5(4(2(3(x1)))))	→	4^#(5(1(x1)))	(78)
5^#(3(1(2(3(x1)))))	→	5^#(x1)	(160)
0^#(5(4(2(3(x1)))))	→	5^#(1(x1))	(76)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
0^#(0(1(2(3(x1)))))	→	0^#(0(x1))	(72)
4^#(0(1(2(3(x1)))))	→	0^#(x1)	(73)
1^#(0(1(2(3(x1)))))	→	0^#(x1)	(159)
0^#(0(1(2(3(x1)))))	→	0^#(x1)	(156)
5^#(0(0(5(3(x1)))))	→	0^#(x1)	(157)
0^#(1(5(3(x1))))	→	5^#(x1)	(99)
0^#(4(5(5(3(x1)))))	→	5^#(x1)	(131)
5^#(0(4(2(3(x1)))))	→	4^#(x1)	(154)
0^#(1(2(3(x1))))	→	0^#(x1)	(95)
0^#(5(2(3(x1))))	→	5^#(3(x1))	(153)
5^#(0(1(2(3(x1)))))	→	5^#(x1)	(62)
0^#(1(5(3(x1))))	→	4^#(5(x1))	(151)
5^#(3(1(2(3(x1)))))	→	1^#(5(x1))	(152)
0^#(4(5(5(3(x1)))))	→	1^#(5(x1))	(149)
1^#(0(1(2(3(x1)))))	→	5^#(1(x1))	(60)
1^#(5(3(5(3(x1)))))	→	1^#(x1)	(58)
0^#(1(2(3(x1))))	→	0^#(x1)	(95)
0^#(5(2(3(x1))))	→	5^#(x1)	(56)
0^#(1(0(2(3(x1)))))	→	0^#(x1)	(54)
5^#(0(4(2(3(x1)))))	→	5^#(3(x1))	(141)

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	1
0	0

· x₁ +

[1(x₁)]

0	0
1	1

· x₁ +

[4(x₁)]

0	0
1	1

· x₁ +

[5(x₁)]

1	0
1	1

· x₁ +

[3(x₁)]

1	1
0	0

· x₁ +

4016

[4^#(x₁)]

0	1
0	0

· x₁ +

8031

[0(x₁)]

0	1
0	1

· x₁ +

[5^#(x₁)]

1	0
0	0

· x₁ +

8032

[2(x₁)]

1	1
1	0

· x₁ +

[1^#(x₁)]

1	0
0	0

· x₁ +

4016

together with the usable rules

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

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

5^#(0(4(2(3(x1)))))	→	4^#(1(5(3(x1))))	(205)
0^#(0(4(2(3(x1)))))	→	4^#(x1)	(138)
0^#(1(0(2(3(x1)))))	→	4^#(1(0(0(x1))))	(203)
0^#(0(5(2(3(x1)))))	→	4^#(0(x1))	(137)
5^#(3(1(2(3(x1)))))	→	1^#(5(3(x1)))	(133)
0^#(4(5(5(3(x1)))))	→	5^#(x1)	(131)
5^#(0(4(2(3(x1)))))	→	5^#(x1)	(199)
0^#(1(0(5(3(x1)))))	→	4^#(x1)	(129)
0^#(1(0(2(3(x1)))))	→	0^#(0(x1))	(128)
5^#(0(1(2(3(x1)))))	→	1^#(x1)	(126)
0^#(1(2(3(3(x1)))))	→	1^#(x1)	(124)
5^#(3(1(2(3(x1)))))	→	5^#(3(x1))	(122)
1^#(0(0(5(3(x1)))))	→	0^#(x1)	(119)
0^#(5(2(1(3(x1)))))	→	5^#(x1)	(193)
1^#(0(1(2(3(x1)))))	→	1^#(x1)	(118)
0^#(1(2(3(3(x1)))))	→	4^#(1(x1))	(116)
0^#(4(5(5(3(x1)))))	→	4^#(1(5(x1)))	(115)
0^#(1(0(2(3(x1)))))	→	1^#(0(0(x1)))	(114)
0^#(0(1(2(3(x1)))))	→	0^#(x1)	(156)
0^#(1(2(5(3(x1)))))	→	1^#(0(x1))	(110)
0^#(5(4(2(3(x1)))))	→	1^#(x1)	(106)
0^#(0(5(2(3(x1)))))	→	0^#(x1)	(182)
0^#(1(2(1(3(x1)))))	→	1^#(x1)	(105)
1^#(0(0(5(3(x1)))))	→	5^#(0(x1))	(181)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
5^#(0(0(5(3(x1)))))	→	5^#(5(0(x1)))	(178)
0^#(1(2(5(3(x1)))))	→	0^#(x1)	(176)
0^#(1(5(3(x1))))	→	5^#(x1)	(99)
5^#(0(4(2(3(x1)))))	→	1^#(5(3(x1)))	(172)
0^#(1(2(3(x1))))	→	0^#(x1)	(95)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
0^#(0(4(2(3(x1)))))	→	4^#(4(x1))	(89)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
4^#(0(1(2(3(x1)))))	→	1^#(0(0(x1)))	(165)
0^#(1(2(3(x1))))	→	1^#(1(x1))	(86)
5^#(0(0(5(3(x1)))))	→	5^#(0(x1))	(164)
0^#(0(1(2(3(x1)))))	→	1^#(0(x1))	(82)
4^#(0(1(2(3(x1)))))	→	0^#(0(x1))	(80)
0^#(5(4(2(3(x1)))))	→	4^#(5(1(x1)))	(78)
5^#(3(1(2(3(x1)))))	→	5^#(x1)	(160)
0^#(5(4(2(3(x1)))))	→	5^#(1(x1))	(76)
0^#(1(2(3(x1))))	→	1^#(x1)	(74)
0^#(0(1(2(3(x1)))))	→	0^#(0(x1))	(72)
4^#(0(1(2(3(x1)))))	→	0^#(x1)	(73)
1^#(0(1(2(3(x1)))))	→	0^#(x1)	(159)
0^#(0(1(2(3(x1)))))	→	0^#(x1)	(156)
5^#(0(0(5(3(x1)))))	→	0^#(x1)	(157)
0^#(1(5(3(x1))))	→	5^#(x1)	(99)
0^#(4(5(5(3(x1)))))	→	5^#(x1)	(131)
5^#(0(4(2(3(x1)))))	→	4^#(x1)	(154)
0^#(1(2(3(x1))))	→	0^#(x1)	(95)
0^#(5(2(3(x1))))	→	5^#(3(x1))	(153)
5^#(0(1(2(3(x1)))))	→	5^#(x1)	(62)
0^#(1(5(3(x1))))	→	4^#(5(x1))	(151)
5^#(3(1(2(3(x1)))))	→	1^#(5(x1))	(152)
0^#(4(5(5(3(x1)))))	→	1^#(5(x1))	(149)
1^#(0(1(2(3(x1)))))	→	5^#(1(x1))	(60)
1^#(5(3(5(3(x1)))))	→	1^#(x1)	(58)
0^#(1(2(3(x1))))	→	0^#(x1)	(95)
0^#(5(2(3(x1))))	→	5^#(x1)	(56)
0^#(1(0(2(3(x1)))))	→	0^#(x1)	(54)
5^#(0(4(2(3(x1)))))	→	5^#(3(x1))	(141)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.