Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212421)

The rewrite relation of the following TRS is considered.

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

0^#(1(2(x1)))	→	0^#(x1)	(49)
5^#(0(1(2(x1))))	→	0^#(2(1(3(3(x1)))))	(50)
0^#(1(2(2(x1))))	→	0^#(2(1(0(2(x1)))))	(51)
0^#(4(5(1(2(x1)))))	→	0^#(2(5(1(1(x1)))))	(52)
0^#(1(2(x1)))	→	0^#(2(2(1(x1))))	(53)
0^#(1(5(2(x1))))	→	0^#(5(x1))	(54)
0^#(1(5(2(x1))))	→	5^#(0(2(1(3(x1)))))	(55)
5^#(0(1(2(x1))))	→	5^#(0(2(1(3(3(x1))))))	(56)
0^#(1(2(4(x1))))	→	0^#(2(2(1(1(x1)))))	(57)
0^#(1(2(2(x1))))	→	0^#(2(2(x1)))	(58)
0^#(2(4(2(x1))))	→	5^#(4(3(2(2(x1)))))	(59)
0^#(1(5(2(x1))))	→	0^#(2(1(3(x1))))	(60)
0^#(1(2(x1)))	→	0^#(2(2(1(4(x1)))))	(61)
0^#(3(5(1(2(x1)))))	→	0^#(x1)	(62)
0^#(2(4(x1)))	→	0^#(2(1(4(3(x1)))))	(63)
0^#(1(2(5(x1))))	→	5^#(2(1(0(x1))))	(64)
0^#(1(2(5(x1))))	→	5^#(5(2(1(0(x1)))))	(65)
0^#(3(1(2(x1))))	→	0^#(2(1(3(2(x1)))))	(66)
0^#(3(1(2(x1))))	→	0^#(3(2(3(1(3(x1))))))	(67)
0^#(1(1(2(5(x1)))))	→	5^#(0(2(5(1(1(x1))))))	(68)
5^#(0(1(2(x1))))	→	0^#(x1)	(69)
0^#(3(1(5(2(x1)))))	→	0^#(3(2(5(1(2(x1))))))	(70)
0^#(4(5(1(2(x1)))))	→	5^#(5(x1))	(71)
0^#(1(5(2(x1))))	→	5^#(x1)	(72)
0^#(1(1(2(5(x1)))))	→	5^#(1(1(x1)))	(73)
0^#(0(4(2(x1))))	→	0^#(2(2(3(4(x1)))))	(74)
0^#(4(0(4(2(x1)))))	→	0^#(0(2(2(x1))))	(75)
5^#(0(1(2(2(x1)))))	→	5^#(0(2(2(1(2(x1))))))	(76)
0^#(4(2(1(4(x1)))))	→	0^#(2(1(4(4(4(x1))))))	(77)
0^#(3(4(2(x1))))	→	0^#(2(2(3(4(x1)))))	(78)
0^#(1(2(5(x1))))	→	0^#(x1)	(79)
0^#(4(1(2(5(x1)))))	→	0^#(2(5(x1)))	(80)
0^#(3(5(1(2(x1)))))	→	5^#(3(2(1(0(x1)))))	(81)
0^#(1(2(x1)))	→	5^#(2(3(x1)))	(82)
0^#(1(2(4(x1))))	→	5^#(5(2(1(x1))))	(83)
0^#(1(2(x1)))	→	0^#(1(3(2(x1))))	(84)
0^#(1(5(2(x1))))	→	5^#(5(0(2(1(3(x1))))))	(85)
0^#(3(5(1(2(x1)))))	→	5^#(5(3(2(1(0(x1))))))	(86)
0^#(4(2(1(2(x1)))))	→	0^#(2(x1))	(87)
0^#(3(1(2(x1))))	→	5^#(0(2(3(x1))))	(88)
0^#(4(2(x1)))	→	0^#(2(3(x1)))	(89)
0^#(4(5(1(2(x1)))))	→	5^#(x1)	(90)
0^#(1(2(4(x1))))	→	5^#(2(1(x1)))	(91)
0^#(4(0(4(2(x1)))))	→	0^#(2(2(x1)))	(92)
0^#(1(4(2(x1))))	→	0^#(5(2(1(4(x1)))))	(93)
0^#(1(2(2(x1))))	→	0^#(2(x1))	(94)
0^#(3(1(2(5(x1)))))	→	0^#(5(x1))	(95)
0^#(4(1(2(2(x1)))))	→	0^#(2(2(3(x1))))	(96)
0^#(1(2(4(x1))))	→	0^#(1(4(2(3(x1)))))	(97)
0^#(4(1(1(2(x1)))))	→	0^#(2(1(x1)))	(98)
0^#(3(4(1(4(x1)))))	→	0^#(5(3(1(4(4(x1))))))	(99)
0^#(1(1(2(5(x1)))))	→	0^#(2(5(1(1(x1)))))	(100)
5^#(0(1(2(2(x1)))))	→	0^#(2(2(1(2(x1)))))	(101)
0^#(2(3(4(2(x1)))))	→	0^#(x1)	(102)
0^#(3(1(2(x1))))	→	0^#(2(5(3(x1))))	(103)
0^#(2(4(2(x1))))	→	0^#(5(4(3(2(2(x1))))))	(104)
5^#(0(1(2(x1))))	→	5^#(0(x1))	(105)
0^#(3(1(2(x1))))	→	0^#(2(2(1(x1))))	(106)
0^#(1(5(2(x1))))	→	0^#(2(2(1(0(5(x1))))))	(107)
0^#(4(2(x1)))	→	5^#(2(x1))	(108)
0^#(1(2(x1)))	→	0^#(5(2(3(x1))))	(109)
0^#(3(1(2(x1))))	→	5^#(3(x1))	(110)
0^#(1(5(2(x1))))	→	5^#(0(2(3(x1))))	(111)
0^#(4(2(x1)))	→	5^#(5(2(x1)))	(112)
0^#(3(1(5(2(x1)))))	→	5^#(1(2(x1)))	(113)
5^#(0(4(4(2(x1)))))	→	0^#(5(2(5(4(4(x1))))))	(114)
0^#(3(1(2(x1))))	→	0^#(2(3(x1)))	(115)
5^#(0(4(4(2(x1)))))	→	5^#(4(4(x1)))	(116)
0^#(1(4(2(x1))))	→	5^#(2(1(4(x1))))	(117)
0^#(4(2(x1)))	→	0^#(5(5(2(x1))))	(118)
0^#(1(5(2(x1))))	→	0^#(2(3(x1)))	(119)
0^#(3(1(2(x1))))	→	0^#(x1)	(120)
0^#(4(5(1(2(x1)))))	→	5^#(1(1(x1)))	(121)
0^#(1(2(x1)))	→	5^#(1(0(5(2(3(x1))))))	(122)
5^#(0(2(4(2(x1)))))	→	5^#(1(4(x1)))	(123)
0^#(4(2(5(2(x1)))))	→	5^#(4(3(2(2(0(x1))))))	(124)
0^#(1(2(x1)))	→	0^#(2(1(3(x1))))	(125)
0^#(1(2(x1)))	→	0^#(2(1(0(x1))))	(126)
0^#(4(5(1(2(x1)))))	→	0^#(5(5(x1)))	(127)
0^#(3(4(1(4(x1)))))	→	5^#(3(1(4(4(x1)))))	(128)
5^#(0(2(4(2(x1)))))	→	0^#(2(2(5(1(4(x1))))))	(129)
0^#(4(2(5(2(x1)))))	→	0^#(x1)	(130)
0^#(1(2(4(x1))))	→	0^#(5(5(2(1(x1)))))	(131)
0^#(0(4(2(x1))))	→	0^#(0(2(2(3(4(x1))))))	(132)
5^#(0(4(4(2(x1)))))	→	5^#(2(5(4(4(x1)))))	(133)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0^#(3(1(2(5(x1)))))	→	0^#(5(x1))	(95)
0^#(4(2(5(2(x1)))))	→	0^#(x1)	(130)
0^#(1(2(2(x1))))	→	0^#(2(x1))	(94)
0^#(4(2(x1)))	→	0^#(2(3(x1)))	(89)
0^#(4(5(1(2(x1)))))	→	5^#(x1)	(90)
0^#(3(1(2(x1))))	→	5^#(0(2(3(x1))))	(88)
0^#(4(2(1(2(x1)))))	→	0^#(2(x1))	(87)
0^#(4(5(1(2(x1)))))	→	0^#(5(5(x1)))	(127)
0^#(4(1(2(5(x1)))))	→	0^#(2(5(x1)))	(80)
0^#(1(2(5(x1))))	→	0^#(x1)	(79)
0^#(3(1(2(x1))))	→	0^#(x1)	(120)
0^#(1(5(2(x1))))	→	0^#(2(3(x1)))	(119)
0^#(3(1(2(x1))))	→	0^#(2(3(x1)))	(115)
0^#(1(5(2(x1))))	→	5^#(x1)	(72)
0^#(4(5(1(2(x1)))))	→	5^#(5(x1))	(71)
5^#(0(1(2(x1))))	→	0^#(x1)	(69)
0^#(1(5(2(x1))))	→	5^#(0(2(3(x1))))	(111)
0^#(3(5(1(2(x1)))))	→	0^#(x1)	(62)
5^#(0(1(2(x1))))	→	5^#(0(x1))	(105)
0^#(2(3(4(2(x1)))))	→	0^#(x1)	(102)
0^#(1(5(2(x1))))	→	0^#(5(x1))	(54)
0^#(1(2(x1)))	→	0^#(x1)	(49)

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
1	1

· x₁ +

[1(x₁)]

0	1
0	0

· x₁ +

[4(x₁)]

1	1
0	0

· x₁ +

[5(x₁)]

x₁ +

[3(x₁)]

1	0
0	0

· x₁ +

[0(x₁)]

1	0
0	0

· x₁ +

[5^#(x₁)]

1	0
1	1

· x₁ +

[2(x₁)]

0	0
1	1

· x₁ +

together with the usable rules

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

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

0^#(3(1(2(5(x1)))))	→	0^#(5(x1))	(95)
0^#(4(2(5(2(x1)))))	→	0^#(x1)	(130)
0^#(1(2(2(x1))))	→	0^#(2(x1))	(94)
0^#(4(2(x1)))	→	0^#(2(3(x1)))	(89)
0^#(4(5(1(2(x1)))))	→	5^#(x1)	(90)
0^#(3(1(2(x1))))	→	5^#(0(2(3(x1))))	(88)
0^#(4(2(1(2(x1)))))	→	0^#(2(x1))	(87)
0^#(4(5(1(2(x1)))))	→	0^#(5(5(x1)))	(127)
0^#(4(1(2(5(x1)))))	→	0^#(2(5(x1)))	(80)
0^#(1(2(5(x1))))	→	0^#(x1)	(79)
0^#(3(1(2(x1))))	→	0^#(x1)	(120)
0^#(1(5(2(x1))))	→	0^#(2(3(x1)))	(119)
0^#(1(5(2(x1))))	→	5^#(x1)	(72)
0^#(4(5(1(2(x1)))))	→	5^#(5(x1))	(71)
5^#(0(1(2(x1))))	→	0^#(x1)	(69)
0^#(1(5(2(x1))))	→	5^#(0(2(3(x1))))	(111)
0^#(3(5(1(2(x1)))))	→	0^#(x1)	(62)
5^#(0(1(2(x1))))	→	5^#(0(x1))	(105)
0^#(2(3(4(2(x1)))))	→	0^#(x1)	(102)
0^#(1(5(2(x1))))	→	0^#(5(x1))	(54)
0^#(1(2(x1)))	→	0^#(x1)	(49)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.