Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z064)

The rewrite relation of the following TRS is considered.

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

→

a(c(b(b(b(a(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.

b^#(a(c(b(b(x1)))))	→	b^#(b(b(a(c(x1)))))	(2)
b^#(a(c(b(b(x1)))))	→	b^#(b(a(c(x1))))	(3)
b^#(a(c(b(b(x1)))))	→	b^#(a(c(x1)))	(4)

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

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

together with the usable rule

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

→

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

(1)

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

b^#(a(c(b(b(x1)))))	→	b^#(b(a(c(x1))))	(3)
b^#(a(c(b(b(x1)))))	→	b^#(a(c(x1)))	(4)

and no rules could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

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

(2)

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

[c(x₁)]

0	0	0
0	0	0
-3	0	0

· x₁ +

-∞

[b(x₁)]

0	0	3
0	0	0
0	0	0

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[b^#(x₁)]

7	8	10
7	8	10
7	8	10

· x₁ +

-∞

together with the usable rule

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

→

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

(1)

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

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

→

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

(2)

could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.