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 ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[b(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	1
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

all of the following rules can be deleted.

a(a(a(x1)))	→	a(b(a(x1)))	(2)
a(a(a(a(x1))))	→	a(a(b(a(a(x1)))))	(4)
a(b(a(a(x1))))	→	b(a(b(b(a(x1)))))	(6)
a(a(b(a(a(x1)))))	→	a(b(a(b(a(b(a(x1)))))))	(10)
a(b(b(a(a(x1)))))	→	b(b(a(b(b(b(a(x1)))))))	(14)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(a(b(a(x1))))	→	a^#(b(x1))	(16)
a^#(a(b(a(x1))))	→	a^#(b(b(a(b(x1)))))	(17)
a^#(a(a(a(a(x1)))))	→	a^#(b(a(a(a(x1)))))	(18)
a^#(a(a(a(a(x1)))))	→	a^#(a(b(a(a(a(x1))))))	(19)
a^#(a(a(a(a(x1)))))	→	a^#(a(a(b(a(a(a(x1)))))))	(20)
a^#(a(a(b(a(x1)))))	→	a^#(b(x1))	(21)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(x1)))	(22)
a^#(a(a(b(a(x1)))))	→	a^#(b(b(a(a(b(x1))))))	(23)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(b(a(a(b(x1)))))))	(24)
a^#(a(b(b(a(x1)))))	→	a^#(b(b(x1)))	(25)
a^#(a(b(b(a(x1)))))	→	a^#(b(b(b(a(b(b(x1)))))))	(26)
a^#(b(a(a(a(x1)))))	→	a^#(b(b(a(a(x1)))))	(27)
a^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(a(x1))))))	(28)
a^#(b(a(b(a(x1)))))	→	a^#(b(x1))	(29)
a^#(b(a(b(a(x1)))))	→	a^#(b(b(b(a(b(x1))))))	(30)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(a(b(a(x1)))))	→	a^#(b(x1))	(29)
a^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(a(x1))))))	(28)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(b(a(a(b(x1)))))))	(24)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(x1)))	(22)
a^#(a(a(b(a(x1)))))	→	a^#(b(x1))	(21)
a^#(a(a(a(a(x1)))))	→	a^#(a(a(b(a(a(a(x1)))))))	(20)
a^#(a(a(a(a(x1)))))	→	a^#(a(b(a(a(a(x1))))))	(19)
a^#(a(a(a(a(x1)))))	→	a^#(b(a(a(a(x1)))))	(18)
a^#(a(b(a(x1))))	→	a^#(b(x1))	(16)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	-∞ · x₁ + 0
[a(x₁)]	=	1 · x₁ + 1
[a^#(x₁)]	=	0 · x₁ + 3

together with the usable rules

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(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(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(b(a(x1)))))	→	b(b(b(b(b(b(b(x1)))))))	(15)

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

a^#(a(a(a(a(x1)))))	→	a^#(a(a(b(a(a(a(x1)))))))	(20)
a^#(a(a(a(a(x1)))))	→	a^#(a(b(a(a(a(x1))))))	(19)
a^#(a(a(a(a(x1)))))	→	a^#(b(a(a(a(x1)))))	(18)

could be deleted.

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	-∞ · x₁ + 3
[a(x₁)]	=	1 · x₁ + 2
[a^#(x₁)]	=	0 · x₁ + 4

together with the usable rules

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(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(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(b(a(x1)))))	→	b(b(b(b(b(b(b(x1)))))))	(15)

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

a^#(a(a(b(a(x1)))))	→	a^#(a(b(b(a(a(b(x1)))))))	(24)
a^#(a(a(b(a(x1)))))	→	a^#(a(b(x1)))	(22)
a^#(a(a(b(a(x1)))))	→	a^#(b(x1))	(21)

could be deleted.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	-1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[a^#(x₁)]	=	1 · x₁ + -16

together with the usable rules

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(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(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(b(a(x1)))))	→	b(b(b(b(b(b(b(x1)))))))	(15)

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

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

→

a^#(b(x1))

(16)

could be deleted.

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	-1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 1
[a^#(x₁)]	=	4 · x₁ + 0

together with the usable rules

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(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(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(b(a(x1)))))	→	b(b(b(b(b(b(b(x1)))))))	(15)

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

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

→

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

(28)

could be deleted.

1.1.1.1.1.1.1.1 Subterm Criterion Processor

We use the projection to multisets

π(a^#)	=	{ 1 }
π(b)	=	{ 1 }

to remove the pairs:

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

→

a^#(b(x1))

(29)

1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.