Certification Problem

Input (TPDB SRS_Standard/Zantema_06/17)

The rewrite relation of the following TRS is considered.

a(a(x1))	→	b(x1)	(1)
a(a(a(x1)))	→	a(b(a(x1)))	(2)
a(b(a(x1)))	→	b(b(b(x1)))	(3)
a(a(a(a(x1))))	→	a(a(b(a(a(x1)))))	(4)
a(a(b(a(x1))))	→	a(b(b(a(b(x1)))))	(5)
a(b(a(a(x1))))	→	b(a(b(b(a(x1)))))	(6)
a(b(b(a(x1))))	→	b(b(b(b(b(x1)))))	(7)
a(a(a(a(a(x1)))))	→	a(a(a(b(a(a(a(x1)))))))	(8)
a(a(a(b(a(x1)))))	→	a(a(b(b(a(a(b(x1)))))))	(9)
a(a(b(a(a(x1)))))	→	a(b(a(b(a(b(a(x1)))))))	(10)
a(a(b(b(a(x1)))))	→	a(b(b(b(a(b(b(x1)))))))	(11)
a(b(a(a(a(x1)))))	→	b(a(a(b(b(a(a(x1)))))))	(12)
a(b(a(b(a(x1)))))	→	b(a(b(b(b(a(b(x1)))))))	(13)
a(b(b(a(a(x1)))))	→	b(b(a(b(b(b(a(x1)))))))	(14)
a(b(b(b(a(x1)))))	→	b(b(b(b(b(b(b(x1)))))))	(15)

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^#(b(a(a(x1))))	→	a^#(b(b(a(x1))))	(16)
a^#(a(a(b(a(x1)))))	→	a^#(b(b(a(a(b(x1))))))	(17)
a^#(b(b(a(a(x1)))))	→	a^#(b(b(b(a(x1)))))	(18)
a^#(a(b(b(a(x1)))))	→	a^#(b(b(b(a(b(b(x1)))))))	(19)
a^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(a(x1))))))	(20)
a^#(a(a(a(x1))))	→	a^#(b(a(a(x1))))	(21)
a^#(b(a(a(a(x1)))))	→	a^#(b(b(a(a(x1)))))	(22)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(b(a(b(a(x1)))))))	(23)
a^#(a(a(a(a(x1)))))	→	a^#(a(b(a(a(a(x1))))))	(24)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(x1)))	(25)
a^#(a(b(b(a(x1)))))	→	a^#(b(b(x1)))	(26)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(b(a(a(b(x1)))))))	(27)
a^#(a(a(a(x1))))	→	a^#(a(b(a(a(x1)))))	(28)
a^#(a(b(a(x1))))	→	a^#(b(b(a(b(x1)))))	(29)
a^#(a(a(a(a(x1)))))	→	a^#(a(a(b(a(a(a(x1)))))))	(30)
a^#(a(a(a(a(x1)))))	→	a^#(b(a(a(a(x1)))))	(31)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(x1)))	(32)
a^#(b(a(b(a(x1)))))	→	a^#(b(b(b(a(b(x1))))))	(33)
a^#(a(a(x1)))	→	a^#(b(a(x1)))	(34)
a^#(a(a(b(a(x1)))))	→	a^#(b(x1))	(35)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(b(a(x1)))))	(36)
a^#(b(a(b(a(x1)))))	→	a^#(b(x1))	(37)
a^#(a(b(a(x1))))	→	a^#(b(x1))	(38)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(a(b(a(x1))))	→	a^#(b(x1))	(38)
a^#(b(a(b(a(x1)))))	→	a^#(b(x1))	(37)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(b(a(b(a(x1)))))))	(23)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(b(a(x1)))))	(36)
a^#(a(a(b(a(x1)))))	→	a^#(b(x1))	(35)
a^#(a(a(x1)))	→	a^#(b(a(x1)))	(34)
a^#(a(a(a(x1))))	→	a^#(b(a(a(x1))))	(21)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(x1)))	(32)
a^#(a(a(a(a(x1)))))	→	a^#(b(a(a(a(x1)))))	(31)
a^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(a(x1))))))	(20)
a^#(a(a(a(a(x1)))))	→	a^#(a(a(b(a(a(a(x1)))))))	(30)
a^#(a(a(a(x1))))	→	a^#(a(b(a(a(x1)))))	(28)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(b(a(a(b(x1)))))))	(27)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(x1)))	(25)
a^#(a(a(a(a(x1)))))	→	a^#(a(b(a(a(a(x1))))))	(24)

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 +

72249

48168

[b(x₁)]

=

1	0
0	0

· x₁ +

24083

1

[a^#(x₁)]

=

1	1
0	0

· x₁ +

0

0

together with the usable rules

a(a(a(a(x1))))	→	a(a(b(a(a(x1)))))	(4)
a(b(b(b(a(x1)))))	→	b(b(b(b(b(b(b(x1)))))))	(15)
a(a(a(a(a(x1)))))	→	a(a(a(b(a(a(a(x1)))))))	(8)
a(a(x1))	→	b(x1)	(1)
a(b(a(x1)))	→	b(b(b(x1)))	(3)
a(a(b(a(x1))))	→	a(b(b(a(b(x1)))))	(5)
a(a(b(a(a(x1)))))	→	a(b(a(b(a(b(a(x1)))))))	(10)
a(b(b(a(x1))))	→	b(b(b(b(b(x1)))))	(7)
a(b(b(a(a(x1)))))	→	b(b(a(b(b(b(a(x1)))))))	(14)
a(b(a(a(a(x1)))))	→	b(a(a(b(b(a(a(x1)))))))	(12)
a(a(b(b(a(x1)))))	→	a(b(b(b(a(b(b(x1)))))))	(11)
a(a(a(b(a(x1)))))	→	a(a(b(b(a(a(b(x1)))))))	(9)
a(b(a(b(a(x1)))))	→	b(a(b(b(b(a(b(x1)))))))	(13)
a(b(a(a(x1))))	→	b(a(b(b(a(x1)))))	(6)
a(a(a(x1)))	→	a(b(a(x1)))	(2)

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

a^#(a(b(a(x1))))	→	a^#(b(x1))	(38)
a^#(b(a(b(a(x1)))))	→	a^#(b(x1))	(37)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(b(a(b(a(x1)))))))	(23)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(b(a(x1)))))	(36)
a^#(a(a(b(a(x1)))))	→	a^#(b(x1))	(35)
a^#(a(a(x1)))	→	a^#(b(a(x1)))	(34)
a^#(a(a(a(x1))))	→	a^#(b(a(a(x1))))	(21)
a^#(a(b(a(a(x1)))))	→	a^#(b(a(x1)))	(32)
a^#(a(a(a(a(x1)))))	→	a^#(b(a(a(a(x1)))))	(31)
a^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(a(x1))))))	(20)
a^#(a(a(a(a(x1)))))	→	a^#(a(a(b(a(a(a(x1)))))))	(30)
a^#(a(a(a(x1))))	→	a^#(a(b(a(a(x1)))))	(28)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(b(a(a(b(x1)))))))	(27)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(x1)))	(25)
a^#(a(a(a(a(x1)))))	→	a^#(a(b(a(a(a(x1))))))	(24)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.