Certification Problem

Input (TPDB SRS_Standard/Trafo_06/un07)

The rewrite relation of the following TRS is considered.

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

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

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	3	3
0	3	3
0	3	3

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[b^#(x₁)]

6	6	9
6	6	9
6	6	9

· x₁ +

-∞

together with the usable rules

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

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

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

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(b(x1)))	→	b^#(a(a(b(x1))))	(9)
b^#(a(a(b(a(b(x1))))))	→	b^#(b(x1))	(10)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]

0	0
-2	-2

· x₁ +

-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞

[b^#(x₁)]

7	9
7	9

· x₁ +

-∞

together with the usable rules

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

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

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

→

b^#(b(x1))

(10)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.