Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z008)

The rewrite relation of the following TRS is considered.

a(s(x1))	→	s(a(x1))	(1)
b(a(b(s(x1))))	→	a(b(s(a(x1))))	(2)
b(a(b(b(x1))))	→	a(b(a(b(x1))))	(3)
a(b(a(a(x1))))	→	b(a(b(a(x1))))	(4)

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.

b^#(a(b(s(x1))))	→	b^#(s(a(x1)))	(5)
b^#(a(b(b(x1))))	→	a^#(b(x1))	(6)
a^#(b(a(a(x1))))	→	a^#(b(a(x1)))	(7)
a^#(s(x1))	→	a^#(x1)	(8)
a^#(b(a(a(x1))))	→	b^#(a(x1))	(9)
b^#(a(b(b(x1))))	→	b^#(a(b(x1)))	(10)
b^#(a(b(s(x1))))	→	a^#(x1)	(11)
b^#(a(b(s(x1))))	→	a^#(b(s(a(x1))))	(12)
a^#(b(a(a(x1))))	→	b^#(a(b(a(x1))))	(13)
b^#(a(b(b(x1))))	→	a^#(b(a(b(x1))))	(14)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(a(b(b(x1))))	→	a^#(b(a(b(x1))))	(14)
a^#(b(a(a(x1))))	→	b^#(a(b(a(x1))))	(13)
b^#(a(b(s(x1))))	→	a^#(x1)	(11)
a^#(b(a(a(x1))))	→	a^#(b(a(x1)))	(7)
b^#(a(b(b(x1))))	→	a^#(b(x1))	(6)
b^#(a(b(b(x1))))	→	b^#(a(b(x1)))	(10)
a^#(b(a(a(x1))))	→	b^#(a(x1))	(9)
a^#(s(x1))	→	a^#(x1)	(8)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 21239
[s(x₁)]	=	x₁ + 7720
[b(x₁)]	=	x₁ + 21239
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

a(b(a(a(x1))))	→	b(a(b(a(x1))))	(4)
a(s(x1))	→	s(a(x1))	(1)
b(a(b(b(x1))))	→	a(b(a(b(x1))))	(3)
b(a(b(s(x1))))	→	a(b(s(a(x1))))	(2)

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

b^#(a(b(s(x1))))	→	a^#(x1)	(11)
a^#(b(a(a(x1))))	→	a^#(b(a(x1)))	(7)
b^#(a(b(b(x1))))	→	a^#(b(x1))	(6)
b^#(a(b(b(x1))))	→	b^#(a(b(x1)))	(10)
a^#(b(a(a(x1))))	→	b^#(a(x1))	(9)
a^#(s(x1))	→	a^#(x1)	(8)

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^#(b(a(a(x1))))	→	b^#(a(b(a(x1))))	(13)
b^#(a(b(b(x1))))	→	a^#(b(a(b(x1))))	(14)

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 +

29924

[s(x₁)]

125

[b(x₁)]

1	0
1	0

· x₁ +

29924

59845

[a^#(x₁)]

1	0
1	1

· x₁ +

[b^#(x₁)]

1	0
1	1

· x₁ +

together with the usable rules

a(b(a(a(x1))))	→	b(a(b(a(x1))))	(4)
a(s(x1))	→	s(a(x1))	(1)
b(a(b(b(x1))))	→	a(b(a(b(x1))))	(3)
b(a(b(s(x1))))	→	a(b(s(a(x1))))	(2)

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

a^#(b(a(a(x1))))	→	b^#(a(b(a(x1))))	(13)
b^#(a(b(b(x1))))	→	a^#(b(a(b(x1))))	(14)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.