Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213407)

The rewrite relation of the following TRS is considered.

0(1(1(x1)))	→	1(0(2(1(x1))))	(1)
0(1(1(x1)))	→	1(1(0(3(2(x1)))))	(2)
0(1(1(x1)))	→	1(4(0(0(2(1(x1))))))	(3)
0(1(4(x1)))	→	1(4(0(3(x1))))	(4)
0(1(4(x1)))	→	4(0(2(1(x1))))	(5)
0(1(4(x1)))	→	4(0(2(1(3(x1)))))	(6)
0(4(1(x1)))	→	0(2(4(1(x1))))	(7)
0(4(1(x1)))	→	4(0(2(1(x1))))	(8)
0(4(1(x1)))	→	4(0(3(1(x1))))	(9)
0(4(1(x1)))	→	1(4(4(0(2(x1)))))	(10)
0(4(1(x1)))	→	2(1(4(0(2(x1)))))	(11)
0(5(4(x1)))	→	1(4(0(0(2(5(x1))))))	(12)
0(5(4(x1)))	→	4(0(2(5(2(5(x1))))))	(13)
0(5(4(x1)))	→	5(0(4(0(3(3(x1))))))	(14)
4(2(1(x1)))	→	4(0(2(1(x1))))	(15)
4(2(1(x1)))	→	1(2(4(0(2(x1)))))	(16)
0(1(0(4(x1))))	→	0(0(2(2(1(4(x1))))))	(17)
0(1(1(2(x1))))	→	5(1(0(2(1(x1)))))	(18)
0(1(1(3(x1))))	→	1(1(5(0(3(x1)))))	(19)
0(1(4(1(x1))))	→	1(0(2(4(1(x1)))))	(20)
0(1(4(3(x1))))	→	1(0(2(4(3(x1)))))	(21)
0(1(4(3(x1))))	→	4(0(2(1(3(x1)))))	(22)
0(4(1(2(x1))))	→	2(0(3(1(4(x1)))))	(23)
0(4(2(1(x1))))	→	0(4(3(0(2(1(x1))))))	(24)
0(5(0(4(x1))))	→	1(5(4(0(0(2(x1))))))	(25)
0(5(1(3(x1))))	→	1(1(5(0(3(x1)))))	(26)
0(5(1(3(x1))))	→	5(3(1(0(3(x1)))))	(27)
0(5(4(1(x1))))	→	4(1(5(4(0(2(x1))))))	(28)
0(5(4(3(x1))))	→	0(2(5(0(3(4(x1))))))	(29)
0(5(4(3(x1))))	→	1(5(3(4(0(3(x1))))))	(30)
1(0(5(4(x1))))	→	1(4(5(0(3(3(x1))))))	(31)
1(0(5(4(x1))))	→	5(5(1(0(2(4(x1))))))	(32)
1(4(2(1(x1))))	→	4(0(2(1(2(1(x1))))))	(33)
4(1(2(1(x1))))	→	4(1(0(2(1(x1)))))	(34)
4(1(2(1(x1))))	→	3(4(0(2(1(1(x1))))))	(35)
4(3(2(1(x1))))	→	0(3(4(0(2(1(x1))))))	(36)
0(0(1(2(3(x1)))))	→	1(0(3(0(0(2(x1))))))	(37)
0(0(5(1(2(x1)))))	→	0(0(2(5(0(1(x1))))))	(38)
0(0(5(1(3(x1)))))	→	4(5(0(0(3(1(x1))))))	(39)
0(1(3(4(2(x1)))))	→	1(3(4(3(0(2(x1))))))	(40)
0(4(5(3(4(x1)))))	→	0(3(2(5(4(4(x1))))))	(41)
0(5(0(2(2(x1)))))	→	0(0(2(5(2(4(x1))))))	(42)
0(5(1(1(3(x1)))))	→	0(3(2(1(1(5(x1))))))	(43)
0(5(1(4(3(x1)))))	→	3(0(2(1(5(4(x1))))))	(44)
0(5(5(1(2(x1)))))	→	1(5(3(0(2(5(x1))))))	(45)
1(0(1(2(4(x1)))))	→	1(1(0(2(3(4(x1))))))	(46)
1(4(2(1(2(x1)))))	→	0(2(2(1(1(4(x1))))))	(47)
4(0(0(5(4(x1)))))	→	5(0(0(4(4(5(x1))))))	(48)
4(2(5(4(1(x1)))))	→	4(4(1(5(3(2(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 144 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

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

→

0^#(1(x1))

(54)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[0^#(x₁)]	=	x₁ + 0
[1(x₁)]	=	21657
[4(x₁)]	=	21657
[5(x₁)]	=	21657
[3(x₁)]	=	1
[4^#(x₁)]	=	0
[0(x₁)]	=	x₁ + 21656
[2(x₁)]	=	1
[1^#(x₁)]	=	0

together with the usable rules

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

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

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

→

0^#(1(x1))

(54)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

4^#(1(2(1(x1))))	→	1^#(1(x1))	(112)
1^#(4(2(1(2(x1)))))	→	1^#(1(4(x1)))	(164)
1^#(4(2(1(2(x1)))))	→	1^#(4(x1))	(150)
1^#(4(2(1(2(x1)))))	→	4^#(x1)	(145)

1.1.2 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[0^#(x₁)]

[1(x₁)]

0	1
0	1

· x₁ +

[4(x₁)]

0	1
0	1

· x₁ +

[5(x₁)]

[3(x₁)]

0	1
0	1

· x₁ +

[4^#(x₁)]

0	1
0	0

· x₁ +

[0(x₁)]

1	0
1	0

· x₁ +

[2(x₁)]

0	1
0	1

· x₁ +

[1^#(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

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

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

4^#(1(2(1(x1))))	→	1^#(1(x1))	(112)
1^#(4(2(1(2(x1)))))	→	1^#(1(4(x1)))	(164)
1^#(4(2(1(2(x1)))))	→	1^#(4(x1))	(150)
1^#(4(2(1(2(x1)))))	→	4^#(x1)	(145)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.