Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/jambox4)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

c^#(c(x1))	→	a^#(b(x1))	(5)
c^#(c(x1))	→	b^#(x1)	(6)
b^#(x1)	→	a^#(a(x1))	(7)
b^#(x1)	→	a^#(x1)	(8)
b^#(b(b(x1)))	→	a^#(c(b(x1)))	(9)
b^#(b(b(x1)))	→	c^#(b(x1))	(10)
a^#(c(a(x1)))	→	a^#(c(c(x1)))	(11)
a^#(c(a(x1)))	→	c^#(c(x1))	(12)
a^#(c(a(x1)))	→	c^#(x1)	(13)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(c(x1))	→	b^#(x1)	(6)
b^#(x1)	→	a^#(x1)	(8)
a^#(c(a(x1)))	→	a^#(c(c(x1)))	(11)
a^#(c(a(x1)))	→	c^#(c(x1))	(12)
a^#(c(a(x1)))	→	c^#(x1)	(13)
b^#(b(b(x1)))	→	a^#(c(b(x1)))	(9)

1.1.1 Reduction Pair Processor

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

[c^#(x₁)]

-∞

-∞	1	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞

-∞	1	0
0	0	-∞
-∞	0	-∞

· x₁

[b^#(x₁)]

-∞

1	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a^#(x₁)]

-∞

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

-∞	-∞	0
0	0	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

0	0	0
0	0	0
1	0	-∞

· x₁

the pair

b^#(x1)

→

a^#(x1)

(8)

could be deleted.

1.1.1.1 Reduction Pair Processor

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

[c^#(x₁)]

-∞

-∞	1	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞

0	1	-∞
0	0	0
0	0	0

· x₁

[b^#(x₁)]

-∞

0	1	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a^#(x₁)]

-∞

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

0	-∞	-∞
0	0	0
0	0	0

· x₁

[b(x₁)]

-∞

0	0	0
0	1	0
0	0	0

· x₁

the pair

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

→

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

(9)

could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

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

(11)

1.1.1.1.1.1 Reduction Pair Processor

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

[a^#(x₁)]

-∞

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞	-∞	0
0	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞	-∞	1
-∞	-∞	0
-∞	-∞	-∞

· x₁

[b(x₁)]

0	1	0
0	0	0
-∞	-∞	-∞

· x₁

the pair

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

→

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

(11)

could be deleted.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.