Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-282)

The rewrite relation of the following TRS is considered.

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

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(x1))	→	a^#(c(x1))	(8)
a^#(b(x1))	→	a^#(a(c(x1)))	(7)

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	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

0	1	1
-∞	0	-∞
0	0	0

· x₁

[c(x₁)]

-∞	0	0
-∞	0	-∞
0	1	1

· x₁

[a(x₁)]

0	0	-∞
-∞	0	-∞
1	0	0

· x₁

the pair

a^#(b(x1))

→

a^#(c(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

[a^#(x₁)]

-∞

0	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

-∞

0	0	-∞
1	0	1
0	0	0

· x₁

[a(x₁)]

-∞

0	0	-∞
0	0	-∞
0	1	0

· x₁

[c(x₁)]

-∞

0	-∞	0
0	-∞	0
1	0	1

· x₁

the pair

a^#(b(x1))

→

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

(7)

could be deleted.

1.1.1.1.1 P is empty

There are no pairs anymore.