Certification Problem

Input (TPDB SRS_Standard/Zantema_06/16)

The rewrite relation of the following TRS is considered.

a(x1)	→	b(x1)	(1)
a(a(x1))	→	a(b(a(x1)))	(2)
a(b(x1))	→	b(b(b(x1)))	(3)
a(a(a(x1)))	→	a(a(b(a(a(x1)))))	(4)
a(a(b(x1)))	→	a(b(b(a(b(x1)))))	(5)
a(b(a(x1)))	→	b(a(b(b(a(x1)))))	(6)
a(b(b(x1)))	→	b(b(b(b(b(x1)))))	(7)
b(a(x1))	→	b(b(b(x1)))	(8)
a(b(a(x1)))	→	a(b(b(a(b(x1)))))	(9)
b(a(a(x1)))	→	b(a(b(b(a(x1)))))	(10)
b(b(a(x1)))	→	b(b(b(b(b(x1)))))	(11)

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.

a^#(a(x1))	→	b^#(a(x1))	(12)
b^#(a(x1))	→	b^#(b(b(x1)))	(13)
a^#(b(a(x1)))	→	a^#(b(b(a(b(x1)))))	(14)
b^#(b(a(x1)))	→	b^#(b(b(x1)))	(15)
a^#(b(a(x1)))	→	a^#(b(x1))	(16)
b^#(b(a(x1)))	→	b^#(b(b(b(b(x1)))))	(17)
a^#(a(b(x1)))	→	b^#(b(a(b(x1))))	(18)
a^#(a(b(x1)))	→	a^#(b(b(a(b(x1)))))	(19)
b^#(b(a(x1)))	→	b^#(b(b(b(x1))))	(20)
b^#(a(x1))	→	b^#(x1)	(21)
b^#(b(a(x1)))	→	b^#(b(x1))	(22)
b^#(a(x1))	→	b^#(b(x1))	(23)
a^#(a(b(x1)))	→	b^#(a(b(x1)))	(24)
b^#(a(a(x1)))	→	a^#(b(b(a(x1))))	(25)
a^#(b(a(x1)))	→	b^#(b(a(b(x1))))	(26)
a^#(b(a(x1)))	→	b^#(x1)	(27)
a^#(a(a(x1)))	→	a^#(b(a(a(x1))))	(28)
a^#(b(a(x1)))	→	b^#(b(a(x1)))	(29)
a^#(x1)	→	b^#(x1)	(30)
b^#(a(a(x1)))	→	b^#(a(x1))	(31)
a^#(a(a(x1)))	→	b^#(a(a(x1)))	(32)
a^#(b(x1))	→	b^#(b(x1))	(33)
a^#(b(b(x1)))	→	b^#(b(b(b(x1))))	(34)
a^#(a(a(x1)))	→	a^#(a(b(a(a(x1)))))	(35)
b^#(a(a(x1)))	→	b^#(a(b(b(a(x1)))))	(36)
a^#(b(x1))	→	b^#(b(b(x1)))	(37)
b^#(a(a(x1)))	→	b^#(b(a(x1)))	(38)
a^#(b(b(x1)))	→	b^#(b(b(b(b(x1)))))	(39)
a^#(b(a(x1)))	→	b^#(a(b(b(a(x1)))))	(40)
a^#(a(x1))	→	a^#(b(a(x1)))	(41)
a^#(b(a(x1)))	→	b^#(a(b(x1)))	(42)
b^#(b(a(x1)))	→	b^#(x1)	(43)
a^#(b(a(x1)))	→	a^#(b(b(a(x1))))	(44)
a^#(b(b(x1)))	→	b^#(b(b(x1)))	(45)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(b(x1)))	→	b^#(b(b(x1)))	(45)
a^#(a(b(x1)))	→	b^#(a(b(x1)))	(24)
a^#(b(a(x1)))	→	a^#(b(b(a(x1))))	(44)
b^#(a(x1))	→	b^#(b(x1))	(23)
b^#(b(a(x1)))	→	b^#(b(x1))	(22)
b^#(b(a(x1)))	→	b^#(x1)	(43)
b^#(a(x1))	→	b^#(x1)	(21)
b^#(b(a(x1)))	→	b^#(b(b(b(x1))))	(20)
a^#(b(a(x1)))	→	b^#(a(b(x1)))	(42)
a^#(a(x1))	→	a^#(b(a(x1)))	(41)
a^#(a(b(x1)))	→	a^#(b(b(a(b(x1)))))	(19)
a^#(b(a(x1)))	→	b^#(a(b(b(a(x1)))))	(40)
a^#(a(b(x1)))	→	b^#(b(a(b(x1))))	(18)
a^#(b(b(x1)))	→	b^#(b(b(b(b(x1)))))	(39)
b^#(a(a(x1)))	→	b^#(b(a(x1)))	(38)
a^#(b(x1))	→	b^#(b(b(x1)))	(37)
b^#(b(a(x1)))	→	b^#(b(b(b(b(x1)))))	(17)
a^#(b(a(x1)))	→	a^#(b(x1))	(16)
b^#(b(a(x1)))	→	b^#(b(b(x1)))	(15)
a^#(a(a(x1)))	→	a^#(a(b(a(a(x1)))))	(35)
b^#(a(a(x1)))	→	b^#(a(b(b(a(x1)))))	(36)
a^#(b(b(x1)))	→	b^#(b(b(b(x1))))	(34)
a^#(b(a(x1)))	→	a^#(b(b(a(b(x1)))))	(14)
a^#(b(x1))	→	b^#(b(x1))	(33)
a^#(a(a(x1)))	→	b^#(a(a(x1)))	(32)
b^#(a(a(x1)))	→	b^#(a(x1))	(31)
a^#(x1)	→	b^#(x1)	(30)
a^#(b(a(x1)))	→	b^#(b(a(x1)))	(29)
a^#(a(a(x1)))	→	a^#(b(a(a(x1))))	(28)
a^#(b(a(x1)))	→	b^#(x1)	(27)
a^#(b(a(x1)))	→	b^#(b(a(b(x1))))	(26)
b^#(a(x1))	→	b^#(b(b(x1)))	(13)
a^#(a(x1))	→	b^#(a(x1))	(12)
b^#(a(a(x1)))	→	a^#(b(b(a(x1))))	(25)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	1
[b(x₁)]	=	0
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	0

together with the usable rules

a(a(a(x1)))	→	a(a(b(a(a(x1)))))	(4)
b(a(x1))	→	b(b(b(x1)))	(8)
a(x1)	→	b(x1)	(1)
a(b(x1))	→	b(b(b(x1)))	(3)
a(a(b(x1)))	→	a(b(b(a(b(x1)))))	(5)
b(a(a(x1)))	→	b(a(b(b(a(x1)))))	(10)
a(b(b(x1)))	→	b(b(b(b(b(x1)))))	(7)
b(b(a(x1)))	→	b(b(b(b(b(x1)))))	(11)
a(b(a(x1)))	→	a(b(b(a(b(x1)))))	(9)
a(b(a(x1)))	→	b(a(b(b(a(x1)))))	(6)
a(a(x1))	→	a(b(a(x1)))	(2)

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

a^#(a(b(x1)))	→	b^#(a(b(x1)))	(24)
a^#(a(x1))	→	a^#(b(a(x1)))	(41)
a^#(a(b(x1)))	→	a^#(b(b(a(b(x1)))))	(19)
a^#(a(b(x1)))	→	b^#(b(a(b(x1))))	(18)
a^#(a(a(x1)))	→	b^#(a(a(x1)))	(32)
a^#(a(a(x1)))	→	a^#(b(a(a(x1))))	(28)
a^#(a(x1))	→	b^#(a(x1))	(12)

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

a^#(a(a(x1)))	→	a^#(a(b(a(a(x1)))))	(35)
b^#(a(x1))	→	b^#(x1)	(21)
b^#(a(x1))	→	b^#(b(x1))	(23)
b^#(a(x1))	→	b^#(b(b(x1)))	(13)
a^#(x1)	→	b^#(x1)	(30)
a^#(b(x1))	→	b^#(b(x1))	(33)
a^#(b(x1))	→	b^#(b(b(x1)))	(37)
b^#(a(a(x1)))	→	b^#(a(x1))	(31)
b^#(a(a(x1)))	→	b^#(b(a(x1)))	(38)
b^#(a(a(x1)))	→	a^#(b(b(a(x1))))	(25)
b^#(a(a(x1)))	→	b^#(a(b(b(a(x1)))))	(36)
a^#(b(b(x1)))	→	b^#(b(b(x1)))	(45)
a^#(b(b(x1)))	→	b^#(b(b(b(x1))))	(34)
a^#(b(b(x1)))	→	b^#(b(b(b(b(x1)))))	(39)
b^#(b(a(x1)))	→	b^#(x1)	(43)
b^#(b(a(x1)))	→	b^#(b(x1))	(22)
b^#(b(a(x1)))	→	b^#(b(b(x1)))	(15)
b^#(b(a(x1)))	→	b^#(b(b(b(x1))))	(20)
b^#(b(a(x1)))	→	b^#(b(b(b(b(x1)))))	(17)
a^#(b(a(x1)))	→	b^#(x1)	(27)
a^#(b(a(x1)))	→	a^#(b(x1))	(16)
a^#(b(a(x1)))	→	b^#(a(b(x1)))	(42)
a^#(b(a(x1)))	→	b^#(b(a(b(x1))))	(26)
a^#(b(a(x1)))	→	a^#(b(b(a(b(x1)))))	(14)
a^#(b(a(x1)))	→	b^#(b(a(x1)))	(29)
a^#(b(a(x1)))	→	a^#(b(b(a(x1))))	(44)
a^#(b(a(x1)))	→	b^#(a(b(b(a(x1)))))	(40)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[b(x₁)]	=	0
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	0

together with the usable rules

a(a(a(x1)))	→	a(a(b(a(a(x1)))))	(4)
b(a(x1))	→	b(b(b(x1)))	(8)
a(x1)	→	b(x1)	(1)
a(b(x1))	→	b(b(b(x1)))	(3)
a(a(b(x1)))	→	a(b(b(a(b(x1)))))	(5)
b(a(a(x1)))	→	b(a(b(b(a(x1)))))	(10)
a(b(b(x1)))	→	b(b(b(b(b(x1)))))	(7)
b(b(a(x1)))	→	b(b(b(b(b(x1)))))	(11)
a(b(a(x1)))	→	a(b(b(a(b(x1)))))	(9)
a(b(a(x1)))	→	b(a(b(b(a(x1)))))	(6)
a(a(x1))	→	a(b(a(x1)))	(2)

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

a^#(a(a(x1)))

→

a^#(a(b(a(a(x1)))))

(35)

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

b^#(a(x1))	→	b^#(x1)	(21)
b^#(a(x1))	→	b^#(b(x1))	(23)
b^#(a(x1))	→	b^#(b(b(x1)))	(13)
a^#(x1)	→	b^#(x1)	(30)
a^#(b(x1))	→	b^#(b(x1))	(33)
a^#(b(x1))	→	b^#(b(b(x1)))	(37)
b^#(a(a(x1)))	→	b^#(a(x1))	(31)
b^#(a(a(x1)))	→	b^#(b(a(x1)))	(38)
b^#(a(a(x1)))	→	a^#(b(b(a(x1))))	(25)
b^#(a(a(x1)))	→	b^#(a(b(b(a(x1)))))	(36)
a^#(b(b(x1)))	→	b^#(b(b(x1)))	(45)
a^#(b(b(x1)))	→	b^#(b(b(b(x1))))	(34)
a^#(b(b(x1)))	→	b^#(b(b(b(b(x1)))))	(39)
b^#(b(a(x1)))	→	b^#(x1)	(43)
b^#(b(a(x1)))	→	b^#(b(x1))	(22)
b^#(b(a(x1)))	→	b^#(b(b(x1)))	(15)
b^#(b(a(x1)))	→	b^#(b(b(b(x1))))	(20)
b^#(b(a(x1)))	→	b^#(b(b(b(b(x1)))))	(17)
a^#(b(a(x1)))	→	b^#(x1)	(27)
a^#(b(a(x1)))	→	a^#(b(x1))	(16)
a^#(b(a(x1)))	→	b^#(a(b(x1)))	(42)
a^#(b(a(x1)))	→	b^#(b(a(b(x1))))	(26)
a^#(b(a(x1)))	→	a^#(b(b(a(b(x1)))))	(14)
a^#(b(a(x1)))	→	b^#(b(a(x1)))	(29)
a^#(b(a(x1)))	→	a^#(b(b(a(x1))))	(44)
a^#(b(a(x1)))	→	b^#(a(b(b(a(x1)))))	(40)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a(x₁)]

0	1
0	1

· x₁ +

[b(x₁)]

0	1
0	0

· x₁ +

[a^#(x₁)]

1	0
0	0

· x₁ +

[b^#(x₁)]

together with the usable rules

a(a(a(x1)))	→	a(a(b(a(a(x1)))))	(4)
b(a(x1))	→	b(b(b(x1)))	(8)
a(x1)	→	b(x1)	(1)
a(b(x1))	→	b(b(b(x1)))	(3)
a(a(b(x1)))	→	a(b(b(a(b(x1)))))	(5)
b(a(a(x1)))	→	b(a(b(b(a(x1)))))	(10)
a(b(b(x1)))	→	b(b(b(b(b(x1)))))	(7)
b(b(a(x1)))	→	b(b(b(b(b(x1)))))	(11)
a(b(a(x1)))	→	a(b(b(a(b(x1)))))	(9)
a(b(a(x1)))	→	b(a(b(b(a(x1)))))	(6)
a(a(x1))	→	a(b(a(x1)))	(2)

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

a^#(b(a(x1)))	→	b^#(x1)	(27)
a^#(b(a(x1)))	→	a^#(b(x1))	(16)
a^#(b(a(x1)))	→	b^#(a(b(x1)))	(42)
a^#(b(a(x1)))	→	b^#(b(a(b(x1))))	(26)
a^#(b(a(x1)))	→	a^#(b(b(a(b(x1)))))	(14)
a^#(b(a(x1)))	→	b^#(b(a(x1)))	(29)
a^#(b(a(x1)))	→	a^#(b(b(a(x1))))	(44)
a^#(b(a(x1)))	→	b^#(a(b(b(a(x1)))))	(40)

could be deleted.

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

b^#(a(x1))	→	b^#(x1)	(21)
b^#(a(x1))	→	b^#(b(x1))	(23)
b^#(a(x1))	→	b^#(b(b(x1)))	(13)
a^#(x1)	→	b^#(x1)	(30)
a^#(b(x1))	→	b^#(b(x1))	(33)
a^#(b(x1))	→	b^#(b(b(x1)))	(37)
b^#(a(a(x1)))	→	b^#(a(x1))	(31)
b^#(a(a(x1)))	→	b^#(b(a(x1)))	(38)
b^#(a(a(x1)))	→	a^#(b(b(a(x1))))	(25)
b^#(a(a(x1)))	→	b^#(a(b(b(a(x1)))))	(36)
a^#(b(b(x1)))	→	b^#(b(b(x1)))	(45)
a^#(b(b(x1)))	→	b^#(b(b(b(x1))))	(34)
a^#(b(b(x1)))	→	b^#(b(b(b(b(x1)))))	(39)
b^#(b(a(x1)))	→	b^#(x1)	(43)
b^#(b(a(x1)))	→	b^#(b(x1))	(22)
b^#(b(a(x1)))	→	b^#(b(b(x1)))	(15)
b^#(b(a(x1)))	→	b^#(b(b(b(x1))))	(20)
b^#(b(a(x1)))	→	b^#(b(b(b(b(x1)))))	(17)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a(x₁)]

1	1
1	1

· x₁ +

[b(x₁)]

0	1
0	0

· x₁ +

[a^#(x₁)]

1	0
0	0

· x₁ +

[b^#(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

a(a(a(x1)))	→	a(a(b(a(a(x1)))))	(4)
b(a(x1))	→	b(b(b(x1)))	(8)
a(x1)	→	b(x1)	(1)
a(b(x1))	→	b(b(b(x1)))	(3)
a(a(b(x1)))	→	a(b(b(a(b(x1)))))	(5)
b(a(a(x1)))	→	b(a(b(b(a(x1)))))	(10)
a(b(b(x1)))	→	b(b(b(b(b(x1)))))	(7)
b(b(a(x1)))	→	b(b(b(b(b(x1)))))	(11)
a(b(a(x1)))	→	a(b(b(a(b(x1)))))	(9)
a(b(a(x1)))	→	b(a(b(b(a(x1)))))	(6)
a(a(x1))	→	a(b(a(x1)))	(2)

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

b^#(a(x1))	→	b^#(x1)	(21)
b^#(a(x1))	→	b^#(b(x1))	(23)
b^#(a(x1))	→	b^#(b(b(x1)))	(13)
a^#(x1)	→	b^#(x1)	(30)
a^#(b(x1))	→	b^#(b(x1))	(33)
a^#(b(x1))	→	b^#(b(b(x1)))	(37)
b^#(a(a(x1)))	→	b^#(a(x1))	(31)
b^#(a(a(x1)))	→	b^#(b(a(x1)))	(38)
b^#(a(a(x1)))	→	a^#(b(b(a(x1))))	(25)
b^#(a(a(x1)))	→	b^#(a(b(b(a(x1)))))	(36)
a^#(b(b(x1)))	→	b^#(b(b(x1)))	(45)
a^#(b(b(x1)))	→	b^#(b(b(b(x1))))	(34)
a^#(b(b(x1)))	→	b^#(b(b(b(b(x1)))))	(39)

could be deleted.

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

b^#(b(a(x1)))	→	b^#(x1)	(43)
b^#(b(a(x1)))	→	b^#(b(x1))	(22)
b^#(b(a(x1)))	→	b^#(b(b(x1)))	(15)
b^#(b(a(x1)))	→	b^#(b(b(b(x1))))	(20)
b^#(b(a(x1)))	→	b^#(b(b(b(b(x1)))))	(17)

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a(x₁)]

1	1
0	1

· x₁ +

5906

35703

[b(x₁)]

1	0
0	0

· x₁ +

[a^#(x₁)]

[b^#(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

a(a(a(x1)))	→	a(a(b(a(a(x1)))))	(4)
b(a(x1))	→	b(b(b(x1)))	(8)
a(x1)	→	b(x1)	(1)
a(b(x1))	→	b(b(b(x1)))	(3)
a(a(b(x1)))	→	a(b(b(a(b(x1)))))	(5)
b(a(a(x1)))	→	b(a(b(b(a(x1)))))	(10)
a(b(b(x1)))	→	b(b(b(b(b(x1)))))	(7)
b(b(a(x1)))	→	b(b(b(b(b(x1)))))	(11)
a(b(a(x1)))	→	a(b(b(a(b(x1)))))	(9)
a(b(a(x1)))	→	b(a(b(b(a(x1)))))	(6)
a(a(x1))	→	a(b(a(x1)))	(2)

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

b^#(b(a(x1)))	→	b^#(x1)	(43)
b^#(b(a(x1)))	→	b^#(b(x1))	(22)
b^#(b(a(x1)))	→	b^#(b(b(x1)))	(15)
b^#(b(a(x1)))	→	b^#(b(b(b(x1))))	(20)
b^#(b(a(x1)))	→	b^#(b(b(b(b(x1)))))	(17)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.