Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-35)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(x1)	→	x1	(1)
a(x1)	→	c(b(b(x1)))	(5)
b(c(a(x1)))	→	a(a(c(x1)))	(6)
c(x1)	→	x1	(4)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(x1)	→	b^#(b(x1))	(8)
b^#(c(a(x1)))	→	a^#(a(c(x1)))	(10)
a^#(x1)	→	b^#(x1)	(9)
b^#(c(a(x1)))	→	a^#(c(x1))	(11)

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
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

-∞	0	0
-∞	-∞	0
-∞	-∞	0

· x₁

[c(x₁)]

-∞

0	-∞	0
1	0	0
0	-∞	0

· x₁

[a(x₁)]

1	0	1
0	0	1
0	-∞	0

· x₁

the pair

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

→

a^#(c(x1))

(11)

could be deleted.

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	1	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b^#(x₁)]

-∞

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

-∞

-∞	0	-∞
0	-∞	-∞
0	-∞	-∞

· x₁

[c(x₁)]

0	-∞	1
-∞	0	0
-∞	0	0

· x₁

[a(x₁)]

0	1	0
-∞	0	-∞
0	1	1

· x₁

the pair

a^#(x1)

→

b^#(x1)

(9)

could be deleted.

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)]

-∞

1	1	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

-∞

-∞	0	-∞
-∞	0	-∞
0	0	-∞

· x₁

[c(x₁)]

-∞

0	0	1
-∞	0	0
-∞	0	0

· x₁

[a(x₁)]

0	1	0
-∞	0	0
0	1	1

· x₁

the pair

a^#(x1)

→

b^#(b(x1))

(8)

could be deleted.

1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pair

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

→

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

(10)

and no rules could be deleted.

1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.