Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-16)

The rewrite relation of the following TRS is considered.

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

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

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

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	2
-2	0

· x₁ +

-∞

[a(x₁)]

0	2
0	0

· x₁ +

-∞

[b^#(x₁)]

13	13
13	13

· x₁ +

-∞

[a^#(x₁)]

11	13
11	13

· x₁ +

-∞

together with the usable rules

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

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

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

→

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

(6)

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

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	2
-2	0

· x₁ +

-∞

[a(x₁)]

0	2
0	0

· x₁ +

-∞

[b^#(x₁)]

7	9
7	9

· x₁ +

-∞

[a^#(x₁)]

8	8
8	8

· x₁ +

-∞

together with the usable rules

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

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

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

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.