Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-499)

The rewrite relation of the following TRS is considered.

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

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^#(a(x1))	→	b^#(a(c(b(x1))))	(4)
a^#(a(x1))	→	a^#(c(b(x1)))	(5)
a^#(a(x1))	→	c^#(b(x1))	(6)
a^#(a(x1))	→	b^#(x1)	(7)
b^#(b(x1))	→	a^#(a(x1))	(8)
b^#(b(x1))	→	a^#(x1)	(9)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(x1))	→	a^#(a(x1))	(8)
a^#(a(x1))	→	b^#(a(c(b(x1))))	(4)
b^#(b(x1))	→	a^#(x1)	(9)
a^#(a(x1))	→	a^#(c(b(x1)))	(5)
a^#(a(x1))	→	b^#(x1)	(7)

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[a^#(x₁)]	=	-1 + 2 · x₁
[b^#(x₁)]	=	-1 + 2 · x₁
[b(x₁)]	=	1 + x₁
[a(x₁)]	=	1 + x₁
[c(x₁)]	=	-1 + x₁

the pairs

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

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

[b^#(x₁)]

-∞

0	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

-∞

0	0	-∞
1	1	0
0	1	0

· x₁

[a^#(x₁)]

-∞

0	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

0	0	0
0	0	0
1	1	0

· x₁

[c(x₁)]

-∞

0	-∞	-∞
0	-∞	-∞
0	-∞	-∞

· x₁

the pair

a^#(a(x1))

→

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

(4)

could be deleted.

1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[a(x₁)]	=	1 · x₁
[b(x₁)]	=	1 · x₁
[c(x₁)]	=	1 · x₁
[b^#(x₁)]	=	1 + 2 · x₁
[a^#(x₁)]	=	2 · x₁

the pair

b^#(b(x1))

→

a^#(a(x1))

(8)

and no rules could be deleted.

1.1.1.1.1.1 P is empty

There are no pairs anymore.