Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212612)

The rewrite relation of the following TRS is considered.

0(0(1(x1)))	→	2(0(3(3(0(1(x1))))))	(1)
0(1(0(x1)))	→	0(1(3(4(0(3(x1))))))	(2)
0(1(0(x1)))	→	2(0(3(0(1(4(x1))))))	(3)
0(1(1(x1)))	→	0(3(1(3(1(x1)))))	(4)
0(1(1(x1)))	→	1(3(0(1(4(x1)))))	(5)
0(1(1(x1)))	→	0(1(3(1(3(1(x1))))))	(6)
0(1(1(x1)))	→	1(3(2(1(3(0(x1))))))	(7)
0(1(1(x1)))	→	1(3(3(1(4(0(x1))))))	(8)
0(1(1(x1)))	→	3(0(3(1(5(1(x1))))))	(9)
0(1(1(x1)))	→	5(0(3(1(5(1(x1))))))	(10)
0(5(0(x1)))	→	3(0(3(5(0(x1)))))	(11)
0(5(0(x1)))	→	3(5(0(0(3(x1)))))	(12)
0(5(0(x1)))	→	5(0(3(0(2(x1)))))	(13)
0(5(0(x1)))	→	5(0(3(3(0(x1)))))	(14)
0(5(0(x1)))	→	4(5(0(3(3(0(x1))))))	(15)
0(5(0(x1)))	→	4(5(0(3(5(0(x1))))))	(16)
0(5(0(x1)))	→	5(3(0(1(3(0(x1))))))	(17)
2(0(0(x1)))	→	0(3(0(3(2(x1)))))	(18)
2(0(0(x1)))	→	0(3(3(0(2(3(x1))))))	(19)
2(0(0(x1)))	→	0(3(5(2(0(3(x1))))))	(20)
5(1(0(x1)))	→	3(5(0(1(4(3(x1))))))	(21)
5(1(0(x1)))	→	3(5(1(4(0(3(x1))))))	(22)
5(1(1(x1)))	→	3(1(5(1(x1))))	(23)
5(1(1(x1)))	→	1(3(1(3(5(x1)))))	(24)
5(1(1(x1)))	→	1(3(3(3(5(1(x1))))))	(25)
5(1(1(x1)))	→	1(3(5(5(1(4(x1))))))	(26)
0(2(0(1(x1))))	→	0(2(3(3(0(1(x1))))))	(27)
0(5(1(0(x1))))	→	0(0(1(3(5(x1)))))	(28)
0(5(4(0(x1))))	→	0(4(5(0(3(x1)))))	(29)
2(0(2(0(x1))))	→	3(0(3(0(2(2(x1))))))	(30)
2(0(4(1(x1))))	→	2(3(0(1(4(4(x1))))))	(31)
2(0(5(0(x1))))	→	0(0(3(5(2(x1)))))	(32)
2(2(4(1(x1))))	→	3(2(4(3(2(1(x1))))))	(33)
5(1(0(1(x1))))	→	0(5(1(4(3(1(x1))))))	(34)
5(1(1(0(x1))))	→	0(5(1(5(1(x1)))))	(35)
5(1(2(0(x1))))	→	3(1(3(5(0(2(x1))))))	(36)
5(1(5(0(x1))))	→	5(3(5(0(1(x1)))))	(37)
5(2(0(1(x1))))	→	5(1(0(3(2(x1)))))	(38)
5(3(1(1(x1))))	→	5(3(1(3(1(5(x1))))))	(39)
5(4(1(1(x1))))	→	5(1(4(1(4(5(x1))))))	(40)
5(5(1(0(x1))))	→	5(0(5(1(3(x1)))))	(41)
5(5(1(1(x1))))	→	5(1(3(5(0(1(x1))))))	(42)
0(2(4(1(0(x1)))))	→	2(4(0(0(1(3(x1))))))	(43)
0(5(5(1(1(x1)))))	→	5(1(3(5(0(1(x1))))))	(44)
2(2(2(4(1(x1)))))	→	1(2(2(1(4(2(x1))))))	(45)
2(5(0(1(1(x1)))))	→	5(1(2(0(1(3(x1))))))	(46)
5(0(2(4(1(x1)))))	→	5(1(4(0(3(2(x1))))))	(47)
5(2(4(1(0(x1)))))	→	0(2(3(4(5(1(x1))))))	(48)
5(3(0(4(1(x1)))))	→	5(3(0(1(4(1(x1))))))	(49)
5(3(4(1(1(x1)))))	→	1(4(3(5(2(1(x1))))))	(50)

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

1(0(0(x1)))	→	1(0(3(3(0(2(x1))))))	(51)
0(1(0(x1)))	→	3(0(4(3(1(0(x1))))))	(52)
0(1(0(x1)))	→	4(1(0(3(0(2(x1))))))	(53)
1(1(0(x1)))	→	1(3(1(3(0(x1)))))	(54)
1(1(0(x1)))	→	4(1(0(3(1(x1)))))	(55)
1(1(0(x1)))	→	1(3(1(3(1(0(x1))))))	(56)
1(1(0(x1)))	→	0(3(1(2(3(1(x1))))))	(57)
1(1(0(x1)))	→	0(4(1(3(3(1(x1))))))	(58)
1(1(0(x1)))	→	1(5(1(3(0(3(x1))))))	(59)
1(1(0(x1)))	→	1(5(1(3(0(5(x1))))))	(60)
0(5(0(x1)))	→	0(5(3(0(3(x1)))))	(61)
0(5(0(x1)))	→	3(0(0(5(3(x1)))))	(62)
0(5(0(x1)))	→	2(0(3(0(5(x1)))))	(63)
0(5(0(x1)))	→	0(3(3(0(5(x1)))))	(64)
0(5(0(x1)))	→	0(3(3(0(5(4(x1))))))	(65)
0(5(0(x1)))	→	0(5(3(0(5(4(x1))))))	(66)
0(5(0(x1)))	→	0(3(1(0(3(5(x1))))))	(67)
0(0(2(x1)))	→	2(3(0(3(0(x1)))))	(68)
0(0(2(x1)))	→	3(2(0(3(3(0(x1))))))	(69)
0(0(2(x1)))	→	3(0(2(5(3(0(x1))))))	(70)
0(1(5(x1)))	→	3(4(1(0(5(3(x1))))))	(71)
0(1(5(x1)))	→	3(0(4(1(5(3(x1))))))	(72)
1(1(5(x1)))	→	1(5(1(3(x1))))	(73)
1(1(5(x1)))	→	5(3(1(3(1(x1)))))	(74)
1(1(5(x1)))	→	1(5(3(3(3(1(x1))))))	(75)
1(1(5(x1)))	→	4(1(5(5(3(1(x1))))))	(76)
1(0(2(0(x1))))	→	1(0(3(3(2(0(x1))))))	(77)
0(1(5(0(x1))))	→	5(3(1(0(0(x1)))))	(78)
0(4(5(0(x1))))	→	3(0(5(4(0(x1)))))	(79)
0(2(0(2(x1))))	→	2(2(0(3(0(3(x1))))))	(80)
1(4(0(2(x1))))	→	4(4(1(0(3(2(x1))))))	(81)
0(5(0(2(x1))))	→	2(5(3(0(0(x1)))))	(82)
1(4(2(2(x1))))	→	1(2(3(4(2(3(x1))))))	(83)
1(0(1(5(x1))))	→	1(3(4(1(5(0(x1))))))	(84)
0(1(1(5(x1))))	→	1(5(1(5(0(x1)))))	(85)
0(2(1(5(x1))))	→	2(0(5(3(1(3(x1))))))	(86)
0(5(1(5(x1))))	→	1(0(5(3(5(x1)))))	(87)
1(0(2(5(x1))))	→	2(3(0(1(5(x1)))))	(88)
1(1(3(5(x1))))	→	5(1(3(1(3(5(x1))))))	(89)
1(1(4(5(x1))))	→	5(4(1(4(1(5(x1))))))	(90)
0(1(5(5(x1))))	→	3(1(5(0(5(x1)))))	(91)
1(1(5(5(x1))))	→	1(0(5(3(1(5(x1))))))	(92)
0(1(4(2(0(x1)))))	→	3(1(0(0(4(2(x1))))))	(93)
1(1(5(5(0(x1)))))	→	1(0(5(3(1(5(x1))))))	(94)
1(4(2(2(2(x1)))))	→	2(4(1(2(2(1(x1))))))	(95)
1(1(0(5(2(x1)))))	→	3(1(0(2(1(5(x1))))))	(96)
1(4(2(0(5(x1)))))	→	2(3(0(4(1(5(x1))))))	(97)
0(1(4(2(5(x1)))))	→	1(5(4(3(2(0(x1))))))	(98)
1(4(0(3(5(x1)))))	→	1(4(1(0(3(5(x1))))))	(99)
1(1(4(3(5(x1)))))	→	1(2(5(3(4(1(x1))))))	(100)

1.1 Closure Under Flat Contexts

Using the flat contexts

{1(☐), 0(☐), 3(☐), 2(☐), 4(☐), 5(☐)}

We obtain the transformed TRS

There are 205 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 1230 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[1₀(x₁)]	=	1 · x₁ + 31
[0₀(x₁)]	=	1 · x₁ + 31
[0₁(x₁)]	=	1 · x₁ + 31
[0₃(x₁)]	=	1 · x₁
[3₃(x₁)]	=	1 · x₁
[3₀(x₁)]	=	1 · x₁ + 3
[0₂(x₁)]	=	1 · x₁ + 18
[2₁(x₁)]	=	1 · x₁ + 21
[2₀(x₁)]	=	1 · x₁ + 24
[2₃(x₁)]	=	1 · x₁ + 9
[2₂(x₁)]	=	1 · x₁ + 27
[0₅(x₁)]	=	1 · x₁
[2₅(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁ + 1
[2₄(x₁)]	=	1 · x₁ + 11
[1₁(x₁)]	=	1 · x₁ + 31
[1₃(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁ + 31
[3₂(x₁)]	=	1 · x₁ + 6
[3₅(x₁)]	=	1 · x₁
[3₄(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁ + 31
[5₃(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁ + 18
[5₅(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁ + 1
[4₁(x₁)]	=	1 · x₁ + 2
[4₀(x₁)]	=	1 · x₁ + 33
[4₃(x₁)]	=	1 · x₁ + 1
[4₂(x₁)]	=	1 · x₁ + 8
[4₅(x₁)]	=	1 · x₁ + 2
[4₄(x₁)]	=	1 · x₁ + 15
[1₂(x₁)]	=	1 · x₁
[1₄(x₁)]	=	1 · x₁ + 16

all of the following rules can be deleted.

There are 1128 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[1₀(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁ + 1
[0₃(x₁)]	=	1 · x₁
[3₃(x₁)]	=	1 · x₁
[3₀(x₁)]	=	1 · x₁
[0₂(x₁)]	=	1 · x₁
[2₄(x₁)]	=	1 · x₁
[0₅(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[3₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[5₃(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁ + 1
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[1₃(x₁)]	=	1 · x₁
[1₄(x₁)]	=	1 · x₁
[4₀(x₁)]	=	1 · x₁
[4₁(x₁)]	=	1 · x₁
[4₅(x₁)]	=	1 · x₁
[2₁(x₁)]	=	1 · x₁
[4₄(x₁)]	=	1 · x₁
[3₂(x₁)]	=	1 · x₁
[2₀(x₁)]	=	1 · x₁
[2₃(x₁)]	=	1 · x₁
[2₂(x₁)]	=	1 · x₁
[2₅(x₁)]	=	1 · x₁
[4₂(x₁)]	=	1 · x₁

all of the following rules can be deleted.

1₀(0₀(0₄(x1)))

→

1₀(0₃(3₃(3₀(0₂(2₄(x1))))))

(292)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₅(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁
[0₃(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[1₀(x₁)]	=	1 · x₁
[3₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁
[5₃(x₁)]	=	1 · x₁
[0₂(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[1₃(x₁)]	=	1 · x₁
[3₃(x₁)]	=	1 · x₁
[1₄(x₁)]	=	1 · x₁ + 1
[4₀(x₁)]	=	1 · x₁
[4₁(x₁)]	=	1 · x₁
[3₀(x₁)]	=	1 · x₁
[4₅(x₁)]	=	1 · x₁
[2₁(x₁)]	=	1 · x₁
[4₄(x₁)]	=	1 · x₁
[3₂(x₁)]	=	1 · x₁
[2₀(x₁)]	=	1 · x₁
[2₃(x₁)]	=	1 · x₁
[2₂(x₁)]	=	1 · x₁
[2₅(x₁)]	=	1 · x₁
[2₄(x₁)]	=	1 · x₁
[4₂(x₁)]	=	1 · x₁

all of the following rules can be deleted.

4₁(1₄(4₀(0₂(2₁(x1)))))	→	4₄(4₄(4₁(1₀(0₃(3₂(2₁(x1)))))))	(1037)
4₁(1₄(4₀(0₂(2₀(x1)))))	→	4₄(4₄(4₁(1₀(0₃(3₂(2₀(x1)))))))	(1038)
4₁(1₄(4₀(0₂(2₃(x1)))))	→	4₄(4₄(4₁(1₀(0₃(3₂(2₃(x1)))))))	(1039)
4₁(1₄(4₀(0₂(2₂(x1)))))	→	4₄(4₄(4₁(1₀(0₃(3₂(2₂(x1)))))))	(1040)
4₁(1₄(4₀(0₂(2₅(x1)))))	→	4₄(4₄(4₁(1₀(0₃(3₂(2₅(x1)))))))	(1041)
4₁(1₄(4₀(0₂(2₄(x1)))))	→	4₄(4₄(4₁(1₀(0₃(3₂(2₄(x1)))))))	(1042)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₅(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁ + 1
[0₃(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[1₀(x₁)]	=	1 · x₁
[3₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁ + 1
[0₀(x₁)]	=	1 · x₁
[5₃(x₁)]	=	1 · x₁
[0₂(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁ + 1
[1₅(x₁)]	=	1 · x₁
[1₃(x₁)]	=	1 · x₁
[3₃(x₁)]	=	1 · x₁
[1₄(x₁)]	=	1 · x₁
[4₀(x₁)]	=	1 · x₁
[4₁(x₁)]	=	1 · x₁
[3₀(x₁)]	=	1 · x₁
[4₅(x₁)]	=	1 · x₁
[2₅(x₁)]	=	1 · x₁ + 1
[3₂(x₁)]	=	1 · x₁
[2₃(x₁)]	=	1 · x₁
[2₁(x₁)]	=	1 · x₁
[2₂(x₁)]	=	1 · x₁
[4₂(x₁)]	=	1 · x₁

all of the following rules can be deleted.

1₁(1₅(5₅(5₁(x1))))	→	1₀(0₅(5₃(3₁(1₅(5₁(x1))))))	(377)
1₁(1₅(5₅(5₀(x1))))	→	1₀(0₅(5₃(3₁(1₅(5₀(x1))))))	(378)
1₁(1₅(5₅(5₃(x1))))	→	1₀(0₅(5₃(3₁(1₅(5₃(x1))))))	(379)
1₁(1₅(5₅(5₂(x1))))	→	1₀(0₅(5₃(3₁(1₅(5₂(x1))))))	(380)
1₁(1₅(5₅(5₅(x1))))	→	1₀(0₅(5₃(3₁(1₅(5₅(x1))))))	(381)
1₁(1₅(5₅(5₄(x1))))	→	1₀(0₅(5₃(3₁(1₅(5₄(x1))))))	(382)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₅(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁
[0₃(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[1₀(x₁)]	=	1 · x₁
[3₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁
[5₃(x₁)]	=	1 · x₁
[0₂(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[1₃(x₁)]	=	1 · x₁
[3₃(x₁)]	=	1 · x₁
[1₄(x₁)]	=	1 · x₁
[4₀(x₁)]	=	1 · x₁
[4₁(x₁)]	=	1 · x₁
[3₀(x₁)]	=	1 · x₁
[4₅(x₁)]	=	1 · x₁
[2₅(x₁)]	=	1 · x₁ + 1
[3₂(x₁)]	=	1 · x₁
[2₃(x₁)]	=	1 · x₁
[2₁(x₁)]	=	1 · x₁
[2₂(x₁)]	=	1 · x₁ + 1
[4₂(x₁)]	=	1 · x₁ + 1

all of the following rules can be deleted.

3₁(1₀(0₂(2₅(5₁(x1)))))	→	3₂(2₃(3₀(0₁(1₅(5₁(x1))))))	(1205)
3₁(1₀(0₂(2₅(5₀(x1)))))	→	3₂(2₃(3₀(0₁(1₅(5₀(x1))))))	(1206)
3₁(1₀(0₂(2₅(5₃(x1)))))	→	3₂(2₃(3₀(0₁(1₅(5₃(x1))))))	(1207)
3₁(1₀(0₂(2₅(5₂(x1)))))	→	3₂(2₃(3₀(0₁(1₅(5₂(x1))))))	(1208)
3₁(1₀(0₂(2₅(5₅(x1)))))	→	3₂(2₃(3₀(0₁(1₅(5₅(x1))))))	(1209)
3₁(1₀(0₂(2₅(5₄(x1)))))	→	3₂(2₃(3₀(0₁(1₅(5₄(x1))))))	(1210)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₅(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁
[0₃(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[1₀(x₁)]	=	1 · x₁
[3₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁
[5₃(x₁)]	=	1 · x₁
[0₂(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[1₃(x₁)]	=	1 · x₁
[3₃(x₁)]	=	1 · x₁
[1₄(x₁)]	=	1 · x₁
[4₀(x₁)]	=	1 · x₁
[4₁(x₁)]	=	1 · x₁
[3₀(x₁)]	=	1 · x₁
[4₅(x₁)]	=	1 · x₁
[2₁(x₁)]	=	1 · x₁ + 1
[2₅(x₁)]	=	1 · x₁
[2₂(x₁)]	=	1 · x₁
[2₃(x₁)]	=	1 · x₁
[4₂(x₁)]	=	1 · x₁

all of the following rules can be deleted.

2₁(1₀(0₂(2₅(5₁(x1)))))	→	2₂(2₃(3₀(0₁(1₅(5₁(x1))))))	(1211)
2₁(1₀(0₂(2₅(5₀(x1)))))	→	2₂(2₃(3₀(0₁(1₅(5₀(x1))))))	(1212)
2₁(1₀(0₂(2₅(5₃(x1)))))	→	2₂(2₃(3₀(0₁(1₅(5₃(x1))))))	(1213)
2₁(1₀(0₂(2₅(5₂(x1)))))	→	2₂(2₃(3₀(0₁(1₅(5₂(x1))))))	(1214)
2₁(1₀(0₂(2₅(5₅(x1)))))	→	2₂(2₃(3₀(0₁(1₅(5₅(x1))))))	(1215)
2₁(1₀(0₂(2₅(5₄(x1)))))	→	2₂(2₃(3₀(0₁(1₅(5₄(x1))))))	(1216)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0₅(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁
[0₃(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[1₀(x₁)]	=	1 · x₁
[3₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁
[5₃(x₁)]	=	1 · x₁
[0₂(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[1₃(x₁)]	=	1 · x₁
[3₃(x₁)]	=	1 · x₁
[1₄(x₁)]	=	1 · x₁
[4₀(x₁)]	=	1 · x₁
[4₁(x₁)]	=	1 · x₁
[3₀(x₁)]	=	1 · x₁
[4₅(x₁)]	=	1 · x₁
[2₅(x₁)]	=	1 · x₁ + 1
[4₂(x₁)]	=	1 · x₁
[2₃(x₁)]	=	1 · x₁
[2₁(x₁)]	=	1 · x₁

all of the following rules can be deleted.

4₁(1₀(0₂(2₅(5₁(x1)))))	→	4₂(2₃(3₀(0₁(1₅(5₁(x1))))))	(1217)
4₁(1₀(0₂(2₅(5₀(x1)))))	→	4₂(2₃(3₀(0₁(1₅(5₀(x1))))))	(1218)
4₁(1₀(0₂(2₅(5₃(x1)))))	→	4₂(2₃(3₀(0₁(1₅(5₃(x1))))))	(1219)
4₁(1₀(0₂(2₅(5₂(x1)))))	→	4₂(2₃(3₀(0₁(1₅(5₂(x1))))))	(1220)
4₁(1₀(0₂(2₅(5₅(x1)))))	→	4₂(2₃(3₀(0₁(1₅(5₅(x1))))))	(1221)
4₁(1₀(0₂(2₅(5₄(x1)))))	→	4₂(2₃(3₀(0₁(1₅(5₄(x1))))))	(1222)

1.1.1.1.1.1.1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 135 ruless (increase limit for explicit display).

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

0₅^#(5₀(0₀(x1)))	→	5₀^#(x1)	(1521)
5₀^#(0₁(1₁(1₅(5₁(x1)))))	→	5₀^#(0₁(x1))	(1588)
5₀^#(0₁(1₁(1₅(5₀(x1)))))	→	5₀^#(0₀(x1))	(1589)
5₀^#(0₁(1₁(1₅(5₀(x1)))))	→	0₀^#(x1)	(1590)
0₀^#(0₁(1₁(1₅(5₁(x1)))))	→	5₀^#(0₁(x1))	(1580)
5₀^#(0₁(1₁(1₅(5₅(x1)))))	→	5₀^#(0₅(x1))	(1593)
5₀^#(0₁(1₁(1₅(5₅(x1)))))	→	0₅^#(x1)	(1594)
0₀^#(0₁(1₁(1₅(5₀(x1)))))	→	5₀^#(0₀(x1))	(1581)
0₀^#(0₁(1₁(1₅(5₀(x1)))))	→	0₀^#(x1)	(1582)
0₀^#(0₁(1₁(1₅(5₅(x1)))))	→	5₀^#(0₅(x1))	(1585)
0₀^#(0₁(1₁(1₅(5₅(x1)))))	→	0₅^#(x1)	(1586)

1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[0₅(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁
[0₃(x₁)]	=	1 · x₁
[3₁(x₁)]	=	1 · x₁
[1₀(x₁)]	=	1 · x₁
[3₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁
[5₃(x₁)]	=	1 · x₁
[0₂(x₁)]	=	1 · x₁
[5₂(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₄(x₁)]	=	1 · x₁
[5₄(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[5₀^#(x₁)]	=	1 · x₁
[0₅^#(x₁)]	=	1 · x₁
[0₀^#(x₁)]	=	1 · x₁

together with the usable rules

0₅(5₀(0₁(x1)))	→	0₃(3₁(1₀(0₃(3₅(5₁(x1))))))	(341)
0₅(5₀(0₀(x1)))	→	0₃(3₁(1₀(0₃(3₅(5₀(x1))))))	(342)
0₅(5₀(0₃(x1)))	→	0₃(3₁(1₀(0₃(3₅(5₃(x1))))))	(343)
0₅(5₀(0₂(x1)))	→	0₃(3₁(1₀(0₃(3₅(5₂(x1))))))	(344)
0₅(5₀(0₅(x1)))	→	0₃(3₁(1₀(0₃(3₅(5₅(x1))))))	(345)
0₅(5₀(0₄(x1)))	→	0₃(3₁(1₀(0₃(3₅(5₄(x1))))))	(346)
5₀(0₁(1₁(1₅(5₁(x1)))))	→	5₁(1₅(5₁(1₅(5₀(0₁(x1))))))	(1115)
5₀(0₁(1₁(1₅(5₀(x1)))))	→	5₁(1₅(5₁(1₅(5₀(0₀(x1))))))	(1116)
5₀(0₁(1₁(1₅(5₃(x1)))))	→	5₁(1₅(5₁(1₅(5₀(0₃(x1))))))	(1117)
5₀(0₁(1₁(1₅(5₂(x1)))))	→	5₁(1₅(5₁(1₅(5₀(0₂(x1))))))	(1118)
5₀(0₁(1₁(1₅(5₅(x1)))))	→	5₁(1₅(5₁(1₅(5₀(0₅(x1))))))	(1119)
5₀(0₁(1₁(1₅(5₄(x1)))))	→	5₁(1₅(5₁(1₅(5₀(0₄(x1))))))	(1120)
5₀(0₅(5₁(1₅(5₁(x1)))))	→	5₁(1₀(0₅(5₃(3₅(5₁(x1))))))	(1187)
5₀(0₅(5₁(1₅(5₀(x1)))))	→	5₁(1₀(0₅(5₃(3₅(5₀(x1))))))	(1188)
5₀(0₅(5₁(1₅(5₃(x1)))))	→	5₁(1₀(0₅(5₃(3₅(5₃(x1))))))	(1189)
5₀(0₅(5₁(1₅(5₂(x1)))))	→	5₁(1₀(0₅(5₃(3₅(5₂(x1))))))	(1190)
5₀(0₅(5₁(1₅(5₅(x1)))))	→	5₁(1₀(0₅(5₃(3₅(5₅(x1))))))	(1191)
5₀(0₅(5₁(1₅(5₄(x1)))))	→	5₁(1₀(0₅(5₃(3₅(5₄(x1))))))	(1192)
0₀(0₁(1₁(1₅(5₁(x1)))))	→	0₁(1₅(5₁(1₅(5₀(0₁(x1))))))	(1091)
0₀(0₁(1₁(1₅(5₀(x1)))))	→	0₁(1₅(5₁(1₅(5₀(0₀(x1))))))	(1092)
0₀(0₁(1₁(1₅(5₃(x1)))))	→	0₁(1₅(5₁(1₅(5₀(0₃(x1))))))	(1093)
0₀(0₁(1₁(1₅(5₂(x1)))))	→	0₁(1₅(5₁(1₅(5₀(0₂(x1))))))	(1094)
0₀(0₁(1₁(1₅(5₅(x1)))))	→	0₁(1₅(5₁(1₅(5₀(0₅(x1))))))	(1095)
0₀(0₁(1₁(1₅(5₄(x1)))))	→	0₁(1₅(5₁(1₅(5₀(0₄(x1))))))	(1096)
0₀(0₅(5₁(1₅(5₁(x1)))))	→	0₁(1₀(0₅(5₃(3₅(5₁(x1))))))	(1163)
0₀(0₅(5₁(1₅(5₀(x1)))))	→	0₁(1₀(0₅(5₃(3₅(5₀(x1))))))	(1164)
0₀(0₅(5₁(1₅(5₃(x1)))))	→	0₁(1₀(0₅(5₃(3₅(5₃(x1))))))	(1165)
0₀(0₅(5₁(1₅(5₂(x1)))))	→	0₁(1₀(0₅(5₃(3₅(5₂(x1))))))	(1166)
0₀(0₅(5₁(1₅(5₅(x1)))))	→	0₁(1₀(0₅(5₃(3₅(5₅(x1))))))	(1167)
0₀(0₅(5₁(1₅(5₄(x1)))))	→	0₁(1₀(0₅(5₃(3₅(5₄(x1))))))	(1168)

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5₀^#(0₁(1₁(1₅(5₁(x1)))))	→	5₀^#(0₁(x1))	(1588)
5₀^#(0₁(1₁(1₅(5₀(x1)))))	→	0₀^#(x1)	(1590)
0₀^#(0₁(1₁(1₅(5₁(x1)))))	→	5₀^#(0₁(x1))	(1580)
5₀^#(0₁(1₁(1₅(5₅(x1)))))	→	0₅^#(x1)	(1594)
0₅^#(5₀(0₀(x1)))	→	5₀^#(x1)	(1521)
0₀^#(0₁(1₁(1₅(5₀(x1)))))	→	0₀^#(x1)	(1582)
0₀^#(0₁(1₁(1₅(5₅(x1)))))	→	0₅^#(x1)	(1586)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[0₁(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 · x₁
[5₀^#(x₁)]	=	1 · x₁
[0₀^#(x₁)]	=	1 · x₁
[0₅^#(x₁)]	=	1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[5₀^#(x₁)]	=	1 · x₁
[0₁(x₁)]	=	1 · x₁
[1₁(x₁)]	=	1 · x₁
[1₅(x₁)]	=	1 · x₁
[5₁(x₁)]	=	1 · x₁
[5₀(x₁)]	=	1 · x₁
[0₀^#(x₁)]	=	1 · x₁
[5₅(x₁)]	=	1 · x₁
[0₅^#(x₁)]	=	1 · x₁
[0₀(x₁)]	=	1 + 1 · x₁

the pair

0₅^#(5₀(0₀(x1)))

→

5₀^#(x1)

(1521)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

5₀^#(0₁(1₁(1₅(5₀(x1))))) → 0₀^#(x1) (1590)

0₀^#(0₁(1₁(1₅(5₁(x1))))) → 5₀^#(0₁(x1)) (1580)

5₀^#(0₁(1₁(1₅(5₁(x1))))) → 5₀^#(0₁(x1)) (1588)

0₀^#(0₁(1₁(1₅(5₀(x1))))) → 0₀^#(x1) (1582)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[5₀^#(x₁)] = -2 + 2 · x₁

[0₁(x₁)] = x₁

[1₁(x₁)] = 2 · x₁

[1₅(x₁)] = x₁

[5₀(x₁)] = 1 + x₁

[0₀^#(x₁)] = 1 + 2 · x₁

[5₁(x₁)] = 1 + 2 · x₁

the pairs

5₀^#(0₁(1₁(1₅(5₀(x1))))) → 0₀^#(x1) (1590)

0₀^#(0₁(1₁(1₅(5₁(x1))))) → 5₀^#(0₁(x1)) (1580)

5₀^#(0₁(1₁(1₅(5₁(x1))))) → 5₀^#(0₁(x1)) (1588)

0₀^#(0₁(1₁(1₅(5₀(x1))))) → 0₀^#(x1) (1582)

could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

[5₀^#(x₁)]	=	-2 + 2 · x₁
[0₁(x₁)]	=	x₁
[1₁(x₁)]	=	2 · x₁
[1₅(x₁)]	=	x₁
[5₀(x₁)]	=	1 + x₁
[0₀^#(x₁)]	=	1 + 2 · x₁
[5₁(x₁)]	=	1 + 2 · x₁

Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212612)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

1.1 Closure Under Flat Contexts

1.1.1 Semantic Labeling

1.1.1.1 Rule Removal

1.1.1.1.1 Rule Removal

1.1.1.1.1.1 Rule Removal

1.1.1.1.1.1.1 Rule Removal

1.1.1.1.1.1.1.1 Rule Removal

1.1.1.1.1.1.1.1.1 Rule Removal

1.1.1.1.1.1.1.1.1.1 Rule Removal

1.1.1.1.1.1.1.1.1.1.1 Dependency Pair Transformation

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty