Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212795)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

2(0(1(0(x1))))	→	0(2(1(0(3(x1)))))	(50)
2(0(0(1(x1))))	→	1(0(0(3(2(x1)))))	(51)
2(0(0(1(x1))))	→	0(2(1(1(0(1(x1))))))	(52)
2(0(0(1(x1))))	→	0(2(0(3(3(1(x1))))))	(53)
2(0(0(1(x1))))	→	2(1(1(0(0(3(x1))))))	(54)
2(2(0(1(x1))))	→	0(3(2(1(2(x1)))))	(55)
2(2(0(1(x1))))	→	0(2(2(1(1(1(x1))))))	(56)
2(4(3(1(x1))))	→	4(3(2(1(1(x1)))))	(57)
2(4(3(1(x1))))	→	5(4(3(3(2(1(x1))))))	(58)
2(4(3(1(x1))))	→	4(4(4(3(2(1(x1))))))	(59)
2(4(3(1(x1))))	→	5(4(4(3(2(1(x1))))))	(60)
2(4(3(1(x1))))	→	5(4(5(3(2(1(x1))))))	(61)
2(4(3(1(x1))))	→	4(5(5(3(2(1(x1))))))	(62)
2(5(3(1(x1))))	→	5(0(1(3(2(x1)))))	(63)
2(5(3(1(x1))))	→	3(2(1(5(4(x1)))))	(64)
2(0(1(3(x1))))	→	0(2(1(0(3(x1)))))	(65)
2(0(4(3(x1))))	→	4(2(0(0(3(x1)))))	(66)
2(0(4(3(x1))))	→	4(2(1(0(3(x1)))))	(67)
2(0(4(3(x1))))	→	4(2(5(0(3(x1)))))	(68)
2(0(4(3(x1))))	→	4(2(0(1(3(x1)))))	(69)
2(0(4(3(x1))))	→	0(3(2(1(4(x1)))))	(70)
2(0(4(3(x1))))	→	0(3(3(4(0(2(x1))))))	(71)
2(0(4(3(x1))))	→	4(1(1(2(0(3(x1))))))	(72)
2(0(4(3(x1))))	→	4(2(5(0(3(3(x1))))))	(73)
2(0(4(3(x1))))	→	0(3(3(3(2(4(x1))))))	(74)
2(0(5(3(x1))))	→	5(2(3(0(3(x1)))))	(75)
2(0(5(3(x1))))	→	0(0(0(3(5(2(x1))))))	(76)
2(0(5(3(x1))))	→	4(5(2(0(0(3(x1))))))	(77)
2(0(5(3(x1))))	→	3(0(3(2(5(5(x1))))))	(78)
2(4(3(2(0(x1)))))	→	5(4(2(2(0(3(x1))))))	(79)
2(0(4(3(0(x1)))))	→	2(4(0(1(3(0(x1))))))	(80)
2(0(5(3(0(x1)))))	→	0(3(5(0(1(2(x1))))))	(81)
2(4(0(0(1(x1)))))	→	2(0(1(5(4(0(x1))))))	(82)
2(5(3(0(1(x1)))))	→	0(2(1(3(5(4(x1))))))	(83)
2(2(0(1(1(x1)))))	→	2(1(0(2(1(1(x1))))))	(84)
2(0(4(3(1(x1)))))	→	4(1(0(3(2(1(x1))))))	(85)
2(0(5(3(1(x1)))))	→	3(2(5(2(0(1(x1))))))	(86)
2(0(0(1(3(x1)))))	→	0(2(2(1(0(3(x1))))))	(87)
2(5(3(1(3(x1)))))	→	2(1(5(3(0(3(x1))))))	(88)
2(0(4(3(3(x1)))))	→	3(2(4(0(0(3(x1))))))	(89)
2(0(0(4(3(x1)))))	→	2(4(4(0(0(3(x1))))))	(90)
2(0(1(4(3(x1)))))	→	4(0(3(2(1(3(x1))))))	(91)
2(5(1(4(3(x1)))))	→	5(4(1(0(3(2(x1))))))	(92)
2(0(2(4(3(x1)))))	→	3(2(4(0(2(3(x1))))))	(93)
2(0(2(4(3(x1)))))	→	3(0(3(2(2(4(x1))))))	(94)
2(0(0(5(3(x1)))))	→	5(4(2(0(3(0(x1))))))	(95)
2(4(0(5(3(x1)))))	→	3(5(4(4(0(2(x1))))))	(96)
2(4(0(5(3(x1)))))	→	0(0(3(4(5(2(x1))))))	(97)
2(4(3(5(3(x1)))))	→	3(3(2(4(5(0(x1))))))	(98)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

2^#(0(1(0(x1))))	→	2^#(1(0(3(x1))))	(99)
2^#(0(0(1(x1))))	→	2^#(x1)	(100)
2^#(0(0(1(x1))))	→	2^#(1(1(0(1(x1)))))	(101)
2^#(0(0(1(x1))))	→	2^#(0(3(3(1(x1)))))	(102)
2^#(0(0(1(x1))))	→	2^#(1(1(0(0(3(x1))))))	(103)
2^#(2(0(1(x1))))	→	2^#(1(2(x1)))	(104)
2^#(2(0(1(x1))))	→	2^#(x1)	(105)
2^#(2(0(1(x1))))	→	2^#(2(1(1(1(x1)))))	(106)
2^#(2(0(1(x1))))	→	2^#(1(1(1(x1))))	(107)
2^#(4(3(1(x1))))	→	2^#(1(1(x1)))	(108)
2^#(4(3(1(x1))))	→	2^#(1(x1))	(109)
2^#(5(3(1(x1))))	→	2^#(x1)	(110)
2^#(5(3(1(x1))))	→	2^#(1(5(4(x1))))	(111)
2^#(0(1(3(x1))))	→	2^#(1(0(3(x1))))	(112)
2^#(0(4(3(x1))))	→	2^#(0(0(3(x1))))	(113)
2^#(0(4(3(x1))))	→	2^#(1(0(3(x1))))	(114)
2^#(0(4(3(x1))))	→	2^#(5(0(3(x1))))	(115)
2^#(0(4(3(x1))))	→	2^#(0(1(3(x1))))	(116)
2^#(0(4(3(x1))))	→	2^#(1(4(x1)))	(117)
2^#(0(4(3(x1))))	→	2^#(x1)	(118)
2^#(0(4(3(x1))))	→	2^#(0(3(x1)))	(119)
2^#(0(4(3(x1))))	→	2^#(5(0(3(3(x1)))))	(120)
2^#(0(4(3(x1))))	→	2^#(4(x1))	(121)
2^#(0(5(3(x1))))	→	2^#(3(0(3(x1))))	(122)
2^#(0(5(3(x1))))	→	2^#(x1)	(123)
2^#(0(5(3(x1))))	→	2^#(0(0(3(x1))))	(124)
2^#(0(5(3(x1))))	→	2^#(5(5(x1)))	(125)
2^#(4(3(2(0(x1)))))	→	2^#(2(0(3(x1))))	(126)
2^#(4(3(2(0(x1)))))	→	2^#(0(3(x1)))	(127)
2^#(0(4(3(0(x1)))))	→	2^#(4(0(1(3(0(x1))))))	(128)
2^#(0(5(3(0(x1)))))	→	2^#(x1)	(129)
2^#(4(0(0(1(x1)))))	→	2^#(0(1(5(4(0(x1))))))	(130)
2^#(5(3(0(1(x1)))))	→	2^#(1(3(5(4(x1)))))	(131)
2^#(2(0(1(1(x1)))))	→	2^#(1(0(2(1(1(x1))))))	(132)
2^#(2(0(1(1(x1)))))	→	2^#(1(1(x1)))	(133)
2^#(0(4(3(1(x1)))))	→	2^#(1(x1))	(134)
2^#(0(5(3(1(x1)))))	→	2^#(5(2(0(1(x1)))))	(135)
2^#(0(5(3(1(x1)))))	→	2^#(0(1(x1)))	(136)
2^#(0(0(1(3(x1)))))	→	2^#(2(1(0(3(x1)))))	(137)
2^#(0(0(1(3(x1)))))	→	2^#(1(0(3(x1))))	(138)
2^#(5(3(1(3(x1)))))	→	2^#(1(5(3(0(3(x1))))))	(139)
2^#(0(4(3(3(x1)))))	→	2^#(4(0(0(3(x1)))))	(140)
2^#(0(0(4(3(x1)))))	→	2^#(4(4(0(0(3(x1))))))	(141)
2^#(0(1(4(3(x1)))))	→	2^#(1(3(x1)))	(142)
2^#(5(1(4(3(x1)))))	→	2^#(x1)	(143)
2^#(0(2(4(3(x1)))))	→	2^#(4(0(2(3(x1)))))	(144)
2^#(0(2(4(3(x1)))))	→	2^#(3(x1))	(145)
2^#(0(2(4(3(x1)))))	→	2^#(2(4(x1)))	(146)
2^#(0(2(4(3(x1)))))	→	2^#(4(x1))	(147)
2^#(0(0(5(3(x1)))))	→	2^#(0(3(0(x1))))	(148)
2^#(4(0(5(3(x1)))))	→	2^#(x1)	(149)
2^#(4(3(5(3(x1)))))	→	2^#(4(5(0(x1))))	(150)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

2^#(2(0(1(x1))))	→	2^#(x1)	(105)
2^#(0(0(1(x1))))	→	2^#(x1)	(100)
2^#(5(3(1(x1))))	→	2^#(x1)	(110)
2^#(0(4(3(x1))))	→	2^#(x1)	(118)
2^#(0(4(3(x1))))	→	2^#(4(x1))	(121)
2^#(4(0(5(3(x1)))))	→	2^#(x1)	(149)
2^#(0(5(3(x1))))	→	2^#(x1)	(123)
2^#(0(5(3(0(x1)))))	→	2^#(x1)	(129)
2^#(0(5(3(1(x1)))))	→	2^#(5(2(0(1(x1)))))	(135)
2^#(5(1(4(3(x1)))))	→	2^#(x1)	(143)
2^#(0(2(4(3(x1)))))	→	2^#(2(4(x1)))	(146)
2^#(0(2(4(3(x1)))))	→	2^#(4(x1))	(147)

1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[2^#(x₁)]

-∞

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[2(x₁)]

-∞

0	0	0
-∞	-∞	0
0	0	0

· x₁

[0(x₁)]

-∞	-∞	-∞
0	0	0
0	0	-∞

· x₁

[1(x₁)]

-∞

0	-∞	0
0	0	0
0	-∞	0

· x₁

[5(x₁)]

-∞

-∞	-∞	-∞
1	-∞	-∞
0	-∞	0

· x₁

[3(x₁)]

0	0	0
0	0	0
0	0	0

· x₁

[4(x₁)]

-∞	-∞	-∞
0	0	0
0	-∞	0

· x₁

the pair

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

→

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

(135)

could be deleted.

1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[2^#(x₁)]

-∞

0	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[2(x₁)]

-∞

0	0	0
0	0	0
0	0	0

· x₁

[0(x₁)]

-∞

0	0	0
0	-∞	0
0	-∞	0

· x₁

[1(x₁)]

-∞

0	-∞	0
0	0	0
0	-∞	0

· x₁

[5(x₁)]

-∞

0	-∞	0
0	-∞	0
-∞	-∞	-∞

· x₁

[3(x₁)]

-∞

0	0	0
0	0	0
0	0	0

· x₁

[4(x₁)]

-∞

0	0	0
0	0	1
0	-∞	0

· x₁

the pairs

2^#(0(4(3(x1))))	→	2^#(x1)	(118)
2^#(4(0(5(3(x1)))))	→	2^#(x1)	(149)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

2^#(0(0(1(x1))))	→	2^#(x1)	(100)
2^#(2(0(1(x1))))	→	2^#(x1)	(105)
2^#(5(3(1(x1))))	→	2^#(x1)	(110)
2^#(0(5(3(x1))))	→	2^#(x1)	(123)
2^#(0(5(3(0(x1)))))	→	2^#(x1)	(129)
2^#(5(1(4(3(x1)))))	→	2^#(x1)	(143)
2^#(0(2(4(3(x1)))))	→	2^#(2(4(x1)))	(146)

1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[2^#(x₁)]

-∞

-∞	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[0(x₁)]

-∞	-∞	-∞
-∞	0	0
-∞	0	-∞

· x₁

[1(x₁)]

-∞

0	-∞	-∞
0	0	-∞
0	-∞	0

· x₁

[2(x₁)]

-∞

-∞	-∞	0
-∞	0	-∞
0	-∞	0

· x₁

[5(x₁)]

0	-∞	-∞
1	-∞	-∞
0	-∞	0

· x₁

[3(x₁)]

0	0	0
-∞	-∞	-∞
-∞	0	0

· x₁

[4(x₁)]

-∞	-∞	-∞
0	-∞	0
0	-∞	0

· x₁

the pairs

2^#(5(3(1(x1))))	→	2^#(x1)	(110)
2^#(0(5(3(x1))))	→	2^#(x1)	(123)
2^#(0(5(3(0(x1)))))	→	2^#(x1)	(129)

could be deleted.

1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[2^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[0(x₁)]

-∞

0	0	0
0	0	0
-∞	0	0

· x₁

[1(x₁)]

-∞

0	0	0
0	0	-∞
0	0	-∞

· x₁

[2(x₁)]

-∞

0	0	0
0	0	0
0	0	-∞

· x₁

[5(x₁)]

-∞

-∞	-∞	-∞
-∞	-∞	0
-∞	-∞	-∞

· x₁

[4(x₁)]

0	0	1
0	0	0
-∞	0	-∞

· x₁

[3(x₁)]

-∞	0	0
-∞	0	0
0	0	0

· x₁

the pair

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

→

2^#(x1)

(143)

could be deleted.

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

2(4(3(1(x1))))	→	4(3(2(1(1(x1)))))	(57)
2(4(3(1(x1))))	→	5(4(3(3(2(1(x1))))))	(58)
2(4(3(1(x1))))	→	4(4(4(3(2(1(x1))))))	(59)
2(4(3(1(x1))))	→	5(4(4(3(2(1(x1))))))	(60)
2(4(3(1(x1))))	→	5(4(5(3(2(1(x1))))))	(61)
2(4(3(1(x1))))	→	4(5(5(3(2(1(x1))))))	(62)
2(4(3(2(0(x1)))))	→	5(4(2(2(0(3(x1))))))	(79)
2(4(0(0(1(x1)))))	→	2(0(1(5(4(0(x1))))))	(82)
2(4(0(5(3(x1)))))	→	3(5(4(4(0(2(x1))))))	(96)
2(4(0(5(3(x1)))))	→	0(0(3(4(5(2(x1))))))	(97)
2(4(3(5(3(x1)))))	→	3(3(2(4(5(0(x1))))))	(98)

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

2^#(0(0(1(x1))))	→	2^#(x1)	(100)
2^#(2(0(1(x1))))	→	2^#(x1)	(105)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

2(4(3(1(x1))))	→	4(3(2(1(1(x1)))))	(57)
2(4(3(1(x1))))	→	5(4(3(3(2(1(x1))))))	(58)
2(4(3(1(x1))))	→	4(4(4(3(2(1(x1))))))	(59)
2(4(3(1(x1))))	→	5(4(4(3(2(1(x1))))))	(60)
2(4(3(1(x1))))	→	5(4(5(3(2(1(x1))))))	(61)
2(4(3(1(x1))))	→	4(5(5(3(2(1(x1))))))	(62)
2(4(3(2(0(x1)))))	→	5(4(2(2(0(3(x1))))))	(79)
2(4(0(0(1(x1)))))	→	2(0(1(5(4(0(x1))))))	(82)
2(4(0(5(3(x1)))))	→	3(5(4(4(0(2(x1))))))	(96)
2(4(0(5(3(x1)))))	→	0(0(3(4(5(2(x1))))))	(97)
2(4(3(5(3(x1)))))	→	3(3(2(4(5(0(x1))))))	(98)

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

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

→

2^#(2(4(x1)))

(146)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.