Certification Problem

Input (TPDB SRS_Standard/Zantema_06/13)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

1/3

all of the following rules can be deleted.

a(a(a(x1)))	→	a(a(x1))	(3)
b(b(x1))	→	a(b(a(x1)))	(4)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

1/3

[b^#(x₁)]

x₁ +

2/3

[a^#(x₁)]

x₁ +

together with the usable rules

a(a(a(b(b(x1)))))	→	b(b(b(x1)))	(1)
b(a(a(a(b(x1)))))	→	a(a(a(b(a(a(a(x1)))))))	(2)

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

b^#(a(a(a(b(x1)))))	→	b^#(a(a(a(x1))))	(5)
b^#(a(a(a(b(x1)))))	→	a^#(x1)	(6)
b^#(a(a(a(b(x1)))))	→	a^#(b(a(a(a(x1)))))	(7)
b^#(a(a(a(b(x1)))))	→	a^#(a(x1))	(8)
b^#(a(a(a(b(x1)))))	→	a^#(a(b(a(a(a(x1))))))	(9)
b^#(a(a(a(b(x1)))))	→	a^#(a(a(x1)))	(10)

and no rules 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

b^#(a(a(a(b(x1)))))	→	a^#(a(a(b(a(a(a(x1)))))))	(11)
a^#(a(a(b(b(x1)))))	→	b^#(b(b(x1)))	(12)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[b(x₁)]

0	0	3
-3	0	0
-3	0	0

· x₁ +

-∞

[a(x₁)]

0	0	0
0	0	0
-3	-3	-3

· x₁ +

-∞

[b^#(x₁)]

16	17	18
16	17	18
16	17	18

· x₁ +

-∞

[a^#(x₁)]

15	17	18
15	17	18
15	17	18

· x₁ +

-∞

together with the usable rules

a(a(a(b(b(x1)))))	→	b(b(b(x1)))	(1)
b(a(a(a(b(x1)))))	→	a(a(a(b(a(a(a(x1)))))))	(2)

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

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

→

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

(12)

could be deleted.

1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

a(a(a(b(b(x1)))))	→	b(b(b(x1)))	(1)
b(a(a(a(b(x1)))))	→	a(a(a(b(a(a(a(x1)))))))	(2)

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

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

→

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

(11)

and no rules could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.