Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abbcaaaa-aaaaabbcaaabbc.srs)

The rewrite relation of the following TRS is considered.

a(b(b(c(a(a(a(a(x1))))))))

→

a(a(a(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))))

(1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(x1))))	(2)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(a(a(a(b(b(c(x1))))))))))	(3)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(b(b(c(x1)))))	(4)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(b(b(c(a(a(a(b(b(c(x1)))))))))))	(5)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(b(b(c(x1))))))	(6)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))	(7)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(a(b(b(c(a(a(a(b(b(c(x1)))))))))))))	(8)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))))	(9)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(x1))))	(2)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(a(a(a(b(b(c(x1))))))))))	(3)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(b(b(c(x1)))))	(4)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(b(b(c(a(a(a(b(b(c(x1)))))))))))	(5)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(b(b(c(x1))))))	(6)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))	(7)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(a(b(b(c(a(a(a(b(b(c(x1)))))))))))))	(8)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))))	(9)

1.1.1 Reduction Pair Processor with Usable Rules

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

[c(x₁)]

0	0
-2	0

· x₁ +

-∞

[b(x₁)]

0	0
0	0

· x₁ +

-∞

[a(x₁)]

0	0
-2	-2

· x₁ +

-∞

[a^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rule

a(b(b(c(a(a(a(a(x1))))))))

→

a(a(a(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))))

(1)

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

a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(b(b(c(x1)))))	(4)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(b(b(c(a(a(a(b(b(c(x1)))))))))))	(5)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(b(b(c(x1))))))	(6)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))	(7)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(a(b(b(c(a(a(a(b(b(c(x1)))))))))))))	(8)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))))	(9)

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(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(x1))))	(2)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(a(a(a(b(b(c(x1))))))))))	(3)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

[c(x₁)]

0	0	0	0
0	0	0	0
-4	0	0	0
-4	-4	0	0

· x₁ +

-∞

[b(x₁)]

0	0	0	0
-4	-4	-4	-4
-4	-4	-4	-4
-4	-4	-4	-4

· x₁ +

-∞

[a(x₁)]

0	0	4
0	0	0
-4	0	0
-4	-4	0

· x₁ +

-∞

[a^#(x₁)]

1	3	3	3
1	3	3	3
1	3	3	3
1	3	3	3

· x₁ +

-∞

together with the usable rule

a(b(b(c(a(a(a(a(x1))))))))

→

a(a(a(a(a(b(b(c(a(a(a(b(b(c(x1))))))))))))))

(1)

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

a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(x1))))	(2)
a^#(b(b(c(a(a(a(a(x1))))))))	→	a^#(b(b(c(a(a(a(b(b(c(x1))))))))))	(3)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.