Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z117)

The rewrite relation of the following TRS is considered.

d(a(x1))	→	b(d(x1))	(1)
b(x1)	→	a(a(a(x1)))	(2)
c(d(c(x1)))	→	a(d(x1))	(3)
b(d(d(x1)))	→	c(c(d(d(c(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(d(x1))	→	d(b(x1))	(5)
b(x1)	→	a(a(a(x1)))	(2)
c(d(c(x1)))	→	d(a(x1))	(6)
d(d(b(x1)))	→	c(d(d(c(c(x1)))))	(7)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(d(x1))	→	d^#(b(x1))	(8)
a^#(d(x1))	→	b^#(x1)	(9)
b^#(x1)	→	a^#(a(a(x1)))	(10)
b^#(x1)	→	a^#(a(x1))	(11)
b^#(x1)	→	a^#(x1)	(12)
c^#(d(c(x1)))	→	d^#(a(x1))	(13)
c^#(d(c(x1)))	→	a^#(x1)	(14)
d^#(d(b(x1)))	→	c^#(d(d(c(c(x1)))))	(15)
d^#(d(b(x1)))	→	d^#(d(c(c(x1))))	(16)
d^#(d(b(x1)))	→	d^#(c(c(x1)))	(17)
d^#(d(b(x1)))	→	c^#(c(x1))	(18)
d^#(d(b(x1)))	→	c^#(x1)	(19)

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pairs

a^#(d(x1))	→	d^#(b(x1))	(8)
b^#(x1)	→	a^#(a(a(x1)))	(10)
b^#(x1)	→	a^#(a(x1))	(11)
b^#(x1)	→	a^#(x1)	(12)
c^#(d(c(x1)))	→	a^#(x1)	(14)
d^#(d(b(x1)))	→	d^#(c(c(x1)))	(17)
d^#(d(b(x1)))	→	c^#(c(x1))	(18)
d^#(d(b(x1)))	→	c^#(x1)	(19)

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

d^#(d(b(x1)))	→	c^#(d(d(c(c(x1)))))	(15)
c^#(d(c(x1)))	→	d^#(a(x1))	(13)
d^#(d(b(x1)))	→	d^#(d(c(c(x1))))	(16)

1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pair

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

→

d^#(d(c(c(x1))))

(16)

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

[d^#(x₁)]

-∞

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[d(x₁)]

-∞	0	0
1	-∞	0
0	0	1

· x₁

[b(x₁)]

0	0	1
0	0	1
-∞	-∞	0

· x₁

[c^#(x₁)]

-∞

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

0	0	0
-∞	-∞	0
-∞	-∞	0

· x₁

[a(x₁)]

-∞	-∞	0
0	0	1
-∞	-∞	0

· x₁

the pair

c^#(d(c(x1)))

→

d^#(a(x1))

(13)

could be deleted.

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[d^#(x₁)]	=	1
[d(x₁)]	=	0
[b(x₁)]	=	1 · x₁
[c^#(x₁)]	=	0
[c(x₁)]	=	0
[a(x₁)]	=	1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

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

→

c^#(d(d(c(c(x1)))))

(15)

could be deleted.

1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.