Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213051)

The rewrite relation of the following TRS is considered.

0(0(1(x1)))	→	0(1(2(0(3(x1)))))	(1)
0(0(1(x1)))	→	0(1(2(0(3(2(x1))))))	(2)
0(0(1(x1)))	→	1(2(0(0(3(2(x1))))))	(3)
0(1(0(x1)))	→	2(0(3(1(2(0(x1))))))	(4)
0(4(1(x1)))	→	4(0(3(1(x1))))	(5)
0(4(1(x1)))	→	2(4(0(3(2(1(x1))))))	(6)
0(4(1(x1)))	→	2(4(0(5(3(1(x1))))))	(7)
0(4(1(x1)))	→	4(2(1(2(0(3(x1))))))	(8)
4(1(0(x1)))	→	1(2(0(4(x1))))	(9)
0(0(1(1(x1))))	→	1(2(0(0(3(1(x1))))))	(10)
0(0(2(1(x1))))	→	2(2(0(3(0(1(x1))))))	(11)
0(0(4(1(x1))))	→	1(4(0(3(0(3(x1))))))	(12)
0(0(5(1(x1))))	→	0(0(3(1(2(5(x1))))))	(13)
0(1(1(0(x1))))	→	1(2(0(0(1(2(x1))))))	(14)
0(1(4(1(x1))))	→	0(1(2(4(1(2(x1))))))	(15)
0(1(5(0(x1))))	→	1(2(5(0(0(3(x1))))))	(16)
0(2(4(1(x1))))	→	4(0(5(3(1(2(x1))))))	(17)
0(4(1(0(x1))))	→	0(4(0(1(2(x1)))))	(18)
0(4(2(1(x1))))	→	4(0(3(2(2(1(x1))))))	(19)
0(4(2(1(x1))))	→	4(0(3(5(1(2(x1))))))	(20)
0(4(4(1(x1))))	→	2(4(0(3(4(1(x1))))))	(21)
0(5(0(1(x1))))	→	0(2(0(3(5(1(x1))))))	(22)
0(5(0(1(x1))))	→	5(2(0(3(0(1(x1))))))	(23)
0(5(1(0(x1))))	→	0(1(2(0(3(5(x1))))))	(24)
0(5(1(0(x1))))	→	2(0(1(3(5(0(x1))))))	(25)
4(0(5(1(x1))))	→	1(4(0(5(3(2(x1))))))	(26)
4(1(0(0(x1))))	→	0(4(1(2(0(x1)))))	(27)
4(1(0(1(x1))))	→	1(1(2(0(4(x1)))))	(28)
4(1(5(0(x1))))	→	1(2(0(5(4(x1)))))	(29)
4(1(5(0(x1))))	→	1(4(0(3(5(x1)))))	(30)
4(3(1(0(x1))))	→	1(4(2(0(3(x1)))))	(31)
4(3(1(0(x1))))	→	2(0(2(1(3(4(x1))))))	(32)
4(3(1(0(x1))))	→	2(1(4(2(0(3(x1))))))	(33)
4(3(1(0(x1))))	→	2(2(4(0(1(3(x1))))))	(34)
5(0(1(0(x1))))	→	0(3(5(1(2(0(x1))))))	(35)
5(0(1(0(x1))))	→	1(5(2(0(3(0(x1))))))	(36)
5(4(1(0(x1))))	→	0(1(2(4(5(2(x1))))))	(37)
0(0(5(5(1(x1)))))	→	1(0(0(3(5(5(x1))))))	(38)
0(1(0(5(0(x1)))))	→	0(1(5(2(0(0(x1))))))	(39)
0(2(5(0(1(x1)))))	→	2(0(3(5(0(1(x1))))))	(40)
0(3(1(0(0(x1)))))	→	0(0(3(0(1(2(x1))))))	(41)
0(4(1(4(1(x1)))))	→	4(4(0(1(2(1(x1))))))	(42)
0(5(5(4(1(x1)))))	→	4(1(0(5(5(3(x1))))))	(43)
4(1(5(0(0(x1)))))	→	1(2(0(4(5(0(x1))))))	(44)
4(1(5(5(0(x1)))))	→	5(1(3(0(5(4(x1))))))	(45)
4(3(1(0(1(x1)))))	→	1(1(2(3(0(4(x1))))))	(46)
4(3(1(1(0(x1)))))	→	1(2(0(1(4(3(x1))))))	(47)
4(3(1(5(0(x1)))))	→	5(1(0(3(2(4(x1))))))	(48)
4(4(1(0(5(x1)))))	→	4(0(5(3(4(1(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 122 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^#(4(1(x1)))	→	4^#(0(3(1(x1))))	(166)
4^#(1(5(5(0(x1)))))	→	4^#(x1)	(167)
0^#(4(4(1(x1))))	→	4^#(0(3(4(1(x1)))))	(113)
4^#(1(5(0(0(x1)))))	→	0^#(4(5(0(x1))))	(110)
4^#(3(1(0(1(x1)))))	→	0^#(4(x1))	(106)
4^#(3(1(1(0(x1)))))	→	0^#(1(4(3(x1))))	(98)
0^#(5(5(4(1(x1)))))	→	4^#(1(0(5(5(3(x1))))))	(156)
4^#(3(1(1(0(x1)))))	→	4^#(3(x1))	(96)
0^#(1(5(0(x1))))	→	0^#(0(3(x1)))	(94)
0^#(0(1(1(x1))))	→	0^#(0(3(1(x1))))	(93)
4^#(1(0(1(x1))))	→	0^#(4(x1))	(150)
4^#(3(1(0(x1))))	→	4^#(x1)	(91)
4^#(1(5(0(x1))))	→	4^#(x1)	(89)
4^#(3(1(0(1(x1)))))	→	4^#(x1)	(88)
0^#(0(4(1(x1))))	→	4^#(0(3(0(3(x1)))))	(87)
4^#(3(1(5(0(x1)))))	→	4^#(x1)	(140)
4^#(1(0(x1)))	→	0^#(4(x1))	(77)
4^#(1(5(0(x1))))	→	4^#(0(3(5(x1))))	(74)
0^#(1(0(5(0(x1)))))	→	0^#(0(x1))	(72)
4^#(1(5(5(0(x1)))))	→	0^#(5(4(x1)))	(71)
0^#(0(5(5(1(x1)))))	→	0^#(0(3(5(5(x1)))))	(135)
4^#(4(1(0(5(x1)))))	→	4^#(1(x1))	(128)
0^#(0(2(1(x1))))	→	0^#(1(x1))	(61)
4^#(1(0(0(x1))))	→	0^#(4(1(2(0(x1)))))	(125)
4^#(1(5(0(0(x1)))))	→	4^#(5(0(x1)))	(55)
4^#(1(0(x1)))	→	4^#(x1)	(123)
4^#(1(5(0(x1))))	→	0^#(5(4(x1)))	(122)
4^#(1(0(1(x1))))	→	4^#(x1)	(51)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

4^#(1(5(5(0(x1)))))	→	4^#(x1)	(167)
4^#(1(5(0(0(x1)))))	→	0^#(4(5(0(x1))))	(110)
4^#(3(1(0(1(x1)))))	→	0^#(4(x1))	(106)
4^#(3(1(1(0(x1)))))	→	0^#(1(4(3(x1))))	(98)
4^#(3(1(1(0(x1)))))	→	4^#(3(x1))	(96)
0^#(1(5(0(x1))))	→	0^#(0(3(x1)))	(94)
0^#(0(1(1(x1))))	→	0^#(0(3(1(x1))))	(93)
4^#(1(0(1(x1))))	→	0^#(4(x1))	(150)
4^#(3(1(0(x1))))	→	4^#(x1)	(91)
4^#(1(5(0(x1))))	→	4^#(x1)	(89)
4^#(3(1(0(1(x1)))))	→	4^#(x1)	(88)
0^#(0(4(1(x1))))	→	4^#(0(3(0(3(x1)))))	(87)
4^#(3(1(5(0(x1)))))	→	4^#(x1)	(140)
4^#(1(0(x1)))	→	0^#(4(x1))	(77)
4^#(1(5(0(x1))))	→	4^#(0(3(5(x1))))	(74)
0^#(1(0(5(0(x1)))))	→	0^#(0(x1))	(72)
4^#(1(5(5(0(x1)))))	→	0^#(5(4(x1)))	(71)
0^#(0(5(5(1(x1)))))	→	0^#(0(3(5(5(x1)))))	(135)
4^#(4(1(0(5(x1)))))	→	4^#(1(x1))	(128)
0^#(0(2(1(x1))))	→	0^#(1(x1))	(61)
4^#(1(5(0(0(x1)))))	→	4^#(5(0(x1)))	(55)
4^#(1(0(x1)))	→	4^#(x1)	(123)
4^#(1(5(0(x1))))	→	0^#(5(4(x1)))	(122)
4^#(1(0(1(x1))))	→	4^#(x1)	(51)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

4^#(1(0(0(x1))))	→	0^#(4(1(2(0(x1)))))	(125)
0^#(4(1(x1)))	→	4^#(0(3(1(x1))))	(166)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

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

together with the usable rules

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

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

4^#(1(0(0(x1))))	→	0^#(4(1(2(0(x1)))))	(125)
0^#(4(1(x1)))	→	4^#(0(3(1(x1))))	(166)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.