Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/96563)

The rewrite relation of the following TRS is considered.

0(1(2(3(x1))))	→	0(2(2(3(x1))))	(1)
1(1(4(4(5(x1)))))	→	5(1(0(0(5(x1)))))	(2)
2(3(3(2(5(x1)))))	→	3(5(4(5(5(x1)))))	(3)
5(1(1(5(3(x1)))))	→	5(1(4(5(3(x1)))))	(4)
0(2(5(1(1(4(x1))))))	→	0(3(2(2(4(5(x1))))))	(5)
1(2(1(4(0(3(x1))))))	→	4(2(5(4(5(3(x1))))))	(6)
1(5(2(0(4(3(x1))))))	→	5(0(1(1(1(x1)))))	(7)
2(2(1(5(4(0(x1))))))	→	2(3(5(4(0(3(x1))))))	(8)
4(0(2(3(1(4(4(x1)))))))	→	4(1(3(0(5(0(5(x1)))))))	(9)
4(3(5(4(3(2(2(2(x1))))))))	→	4(1(2(3(4(1(4(x1)))))))	(10)
0(2(2(2(3(4(3(3(3(x1)))))))))	→	0(5(4(0(5(3(5(5(3(x1)))))))))	(11)
4(5(3(4(1(1(0(5(2(x1)))))))))	→	4(4(3(1(3(2(0(5(2(x1)))))))))	(12)
3(3(4(1(0(4(5(1(5(2(x1))))))))))	→	3(3(4(1(5(0(3(0(2(4(x1))))))))))	(13)
2(4(5(4(0(5(5(1(2(1(4(2(x1))))))))))))	→	4(1(0(1(1(3(3(0(1(4(2(x1)))))))))))	(14)
1(0(0(4(5(4(3(4(5(1(5(4(0(x1)))))))))))))	→	5(3(3(1(2(1(1(0(3(1(3(x1)))))))))))	(15)
1(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	4(4(4(3(2(0(1(3(1(2(2(4(5(x1)))))))))))))	(16)
4(1(5(4(5(3(3(3(1(2(0(2(0(x1)))))))))))))	→	4(1(4(3(4(4(0(1(3(1(1(5(x1))))))))))))	(17)
2(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	5(5(5(2(0(2(5(0(0(4(3(2(1(4(3(3(1(x1)))))))))))))))))	(18)
1(5(1(1(3(0(3(3(3(3(1(1(5(1(5(2(5(2(x1))))))))))))))))))	→	3(5(4(2(0(5(1(3(4(1(1(2(5(2(4(1(5(2(x1))))))))))))))))))	(19)
2(4(3(3(4(2(3(5(4(0(4(0(2(3(0(3(3(0(x1))))))))))))))))))	→	2(1(3(0(4(4(1(5(2(1(0(3(0(5(0(1(5(x1)))))))))))))))))	(20)
2(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	2(1(5(0(0(1(5(2(2(2(3(1(0(5(0(4(3(3(x1))))))))))))))))))	(21)
2(3(5(1(3(0(0(5(0(2(2(4(5(4(0(3(1(4(2(x1)))))))))))))))))))	→	5(2(0(2(0(3(2(3(5(2(2(4(1(3(5(5(1(2(x1))))))))))))))))))	(22)
3(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	3(2(0(1(5(0(1(5(5(1(2(3(0(2(4(2(5(2(2(4(x1))))))))))))))))))))	(23)
3(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	3(1(3(3(0(5(4(2(4(1(2(3(5(0(3(4(5(2(4(3(x1))))))))))))))))))))	(24)
4(5(1(4(5(3(1(5(5(3(5(0(2(1(2(4(4(5(1(0(4(x1)))))))))))))))))))))	→	4(4(0(2(1(5(2(0(4(5(2(2(3(5(5(2(2(0(3(5(4(x1)))))))))))))))))))))	(25)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0(x₁)]	=	1 · x₁ + 5
[1(x₁)]	=	1 · x₁ + 6
[2(x₁)]	=	1 · x₁ + 5
[3(x₁)]	=	1 · x₁ + 5
[4(x₁)]	=	1 · x₁ + 5
[5(x₁)]	=	1 · x₁ + 5

all of the following rules can be deleted.

0(1(2(3(x1))))	→	0(2(2(3(x1))))	(1)
1(1(4(4(5(x1)))))	→	5(1(0(0(5(x1)))))	(2)
5(1(1(5(3(x1)))))	→	5(1(4(5(3(x1)))))	(4)
0(2(5(1(1(4(x1))))))	→	0(3(2(2(4(5(x1))))))	(5)
1(2(1(4(0(3(x1))))))	→	4(2(5(4(5(3(x1))))))	(6)
1(5(2(0(4(3(x1))))))	→	5(0(1(1(1(x1)))))	(7)
2(2(1(5(4(0(x1))))))	→	2(3(5(4(0(3(x1))))))	(8)
4(3(5(4(3(2(2(2(x1))))))))	→	4(1(2(3(4(1(4(x1)))))))	(10)
4(5(3(4(1(1(0(5(2(x1)))))))))	→	4(4(3(1(3(2(0(5(2(x1)))))))))	(12)
3(3(4(1(0(4(5(1(5(2(x1))))))))))	→	3(3(4(1(5(0(3(0(2(4(x1))))))))))	(13)
2(4(5(4(0(5(5(1(2(1(4(2(x1))))))))))))	→	4(1(0(1(1(3(3(0(1(4(2(x1)))))))))))	(14)
1(0(0(4(5(4(3(4(5(1(5(4(0(x1)))))))))))))	→	5(3(3(1(2(1(1(0(3(1(3(x1)))))))))))	(15)
4(1(5(4(5(3(3(3(1(2(0(2(0(x1)))))))))))))	→	4(1(4(3(4(4(0(1(3(1(1(5(x1))))))))))))	(17)
1(5(1(1(3(0(3(3(3(3(1(1(5(1(5(2(5(2(x1))))))))))))))))))	→	3(5(4(2(0(5(1(3(4(1(1(2(5(2(4(1(5(2(x1))))))))))))))))))	(19)
2(4(3(3(4(2(3(5(4(0(4(0(2(3(0(3(3(0(x1))))))))))))))))))	→	2(1(3(0(4(4(1(5(2(1(0(3(0(5(0(1(5(x1)))))))))))))))))	(20)
2(3(5(1(3(0(0(5(0(2(2(4(5(4(0(3(1(4(2(x1)))))))))))))))))))	→	5(2(0(2(0(3(2(3(5(2(2(4(1(3(5(5(1(2(x1))))))))))))))))))	(22)
4(5(1(4(5(3(1(5(5(3(5(0(2(1(2(4(4(5(1(0(4(x1)))))))))))))))))))))	→	4(4(0(2(1(5(2(0(4(5(2(2(3(5(5(2(2(0(3(5(4(x1)))))))))))))))))))))	(25)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

2^#(3(3(2(5(x1)))))	→	3^#(5(4(5(5(x1)))))	(26)
2^#(3(3(2(5(x1)))))	→	4^#(5(5(x1)))	(27)
4^#(0(2(3(1(4(4(x1)))))))	→	4^#(1(3(0(5(0(5(x1)))))))	(28)
4^#(0(2(3(1(4(4(x1)))))))	→	1^#(3(0(5(0(5(x1))))))	(29)
4^#(0(2(3(1(4(4(x1)))))))	→	3^#(0(5(0(5(x1)))))	(30)
4^#(0(2(3(1(4(4(x1)))))))	→	0^#(5(0(5(x1))))	(31)
4^#(0(2(3(1(4(4(x1)))))))	→	0^#(5(x1))	(32)
0^#(2(2(2(3(4(3(3(3(x1)))))))))	→	0^#(5(4(0(5(3(5(5(3(x1)))))))))	(33)
0^#(2(2(2(3(4(3(3(3(x1)))))))))	→	4^#(0(5(3(5(5(3(x1)))))))	(34)
0^#(2(2(2(3(4(3(3(3(x1)))))))))	→	0^#(5(3(5(5(3(x1))))))	(35)
0^#(2(2(2(3(4(3(3(3(x1)))))))))	→	3^#(5(5(3(x1))))	(36)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	4^#(4(4(3(2(0(1(3(1(2(2(4(5(x1)))))))))))))	(37)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	4^#(4(3(2(0(1(3(1(2(2(4(5(x1))))))))))))	(38)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	4^#(3(2(0(1(3(1(2(2(4(5(x1)))))))))))	(39)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	3^#(2(0(1(3(1(2(2(4(5(x1))))))))))	(40)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	2^#(0(1(3(1(2(2(4(5(x1)))))))))	(41)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	0^#(1(3(1(2(2(4(5(x1))))))))	(42)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	1^#(3(1(2(2(4(5(x1)))))))	(43)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	3^#(1(2(2(4(5(x1))))))	(44)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	1^#(2(2(4(5(x1)))))	(45)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	2^#(2(4(5(x1))))	(46)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	2^#(4(5(x1)))	(47)
1^#(2(2(1(2(3(2(4(4(0(0(3(2(x1)))))))))))))	→	4^#(5(x1))	(48)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	2^#(0(2(5(0(0(4(3(2(1(4(3(3(1(x1))))))))))))))	(49)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	0^#(2(5(0(0(4(3(2(1(4(3(3(1(x1)))))))))))))	(50)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	2^#(5(0(0(4(3(2(1(4(3(3(1(x1))))))))))))	(51)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	0^#(0(4(3(2(1(4(3(3(1(x1))))))))))	(52)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	0^#(4(3(2(1(4(3(3(1(x1)))))))))	(53)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	4^#(3(2(1(4(3(3(1(x1))))))))	(54)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	3^#(2(1(4(3(3(1(x1)))))))	(55)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	2^#(1(4(3(3(1(x1))))))	(56)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	1^#(4(3(3(1(x1)))))	(57)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	4^#(3(3(1(x1))))	(58)
2^#(2(3(3(0(2(0(2(5(0(5(5(5(1(0(3(1(x1)))))))))))))))))	→	3^#(3(1(x1)))	(59)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	2^#(1(5(0(0(1(5(2(2(2(3(1(0(5(0(4(3(3(x1))))))))))))))))))	(60)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	1^#(5(0(0(1(5(2(2(2(3(1(0(5(0(4(3(3(x1)))))))))))))))))	(61)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	0^#(0(1(5(2(2(2(3(1(0(5(0(4(3(3(x1)))))))))))))))	(62)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	0^#(1(5(2(2(2(3(1(0(5(0(4(3(3(x1))))))))))))))	(63)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	1^#(5(2(2(2(3(1(0(5(0(4(3(3(x1)))))))))))))	(64)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	2^#(2(2(3(1(0(5(0(4(3(3(x1)))))))))))	(65)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	2^#(2(3(1(0(5(0(4(3(3(x1))))))))))	(66)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	2^#(3(1(0(5(0(4(3(3(x1)))))))))	(67)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	3^#(1(0(5(0(4(3(3(x1))))))))	(68)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	1^#(0(5(0(4(3(3(x1)))))))	(69)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	0^#(5(0(4(3(3(x1))))))	(70)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	0^#(4(3(3(x1))))	(71)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	4^#(3(3(x1)))	(72)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	3^#(3(x1))	(73)
2^#(5(5(1(2(3(5(0(3(0(0(3(3(1(0(1(3(2(x1))))))))))))))))))	→	3^#(x1)	(74)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	3^#(2(0(1(5(0(1(5(5(1(2(3(0(2(4(2(5(2(2(4(x1))))))))))))))))))))	(75)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	2^#(0(1(5(0(1(5(5(1(2(3(0(2(4(2(5(2(2(4(x1)))))))))))))))))))	(76)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	0^#(1(5(0(1(5(5(1(2(3(0(2(4(2(5(2(2(4(x1))))))))))))))))))	(77)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	1^#(5(0(1(5(5(1(2(3(0(2(4(2(5(2(2(4(x1)))))))))))))))))	(78)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	0^#(1(5(5(1(2(3(0(2(4(2(5(2(2(4(x1)))))))))))))))	(79)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	1^#(5(5(1(2(3(0(2(4(2(5(2(2(4(x1))))))))))))))	(80)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	1^#(2(3(0(2(4(2(5(2(2(4(x1)))))))))))	(81)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	2^#(3(0(2(4(2(5(2(2(4(x1))))))))))	(82)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	3^#(0(2(4(2(5(2(2(4(x1)))))))))	(83)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	0^#(2(4(2(5(2(2(4(x1))))))))	(84)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	2^#(4(2(5(2(2(4(x1)))))))	(85)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	4^#(2(5(2(2(4(x1))))))	(86)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	2^#(5(2(2(4(x1)))))	(87)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	2^#(2(4(x1)))	(88)
3^#(2(3(4(1(2(5(5(0(1(2(3(4(0(5(1(4(4(5(4(x1))))))))))))))))))))	→	2^#(4(x1))	(89)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	3^#(1(3(3(0(5(4(2(4(1(2(3(5(0(3(4(5(2(4(3(x1))))))))))))))))))))	(90)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	1^#(3(3(0(5(4(2(4(1(2(3(5(0(3(4(5(2(4(3(x1)))))))))))))))))))	(91)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	3^#(3(0(5(4(2(4(1(2(3(5(0(3(4(5(2(4(3(x1))))))))))))))))))	(92)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	3^#(0(5(4(2(4(1(2(3(5(0(3(4(5(2(4(3(x1)))))))))))))))))	(93)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	0^#(5(4(2(4(1(2(3(5(0(3(4(5(2(4(3(x1))))))))))))))))	(94)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	4^#(2(4(1(2(3(5(0(3(4(5(2(4(3(x1))))))))))))))	(95)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	2^#(4(1(2(3(5(0(3(4(5(2(4(3(x1)))))))))))))	(96)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	4^#(1(2(3(5(0(3(4(5(2(4(3(x1))))))))))))	(97)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	1^#(2(3(5(0(3(4(5(2(4(3(x1)))))))))))	(98)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	2^#(3(5(0(3(4(5(2(4(3(x1))))))))))	(99)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	3^#(5(0(3(4(5(2(4(3(x1)))))))))	(100)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	0^#(3(4(5(2(4(3(x1)))))))	(101)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	3^#(4(5(2(4(3(x1))))))	(102)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	4^#(5(2(4(3(x1)))))	(103)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	2^#(4(3(x1)))	(104)
3^#(5(5(2(0(4(4(5(5(0(3(5(2(1(2(1(0(2(2(3(x1))))))))))))))))))))	→	4^#(3(x1))	(105)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.