Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/jambox3)

The rewrite relation of the following TRS is considered.

a(a(c(x1)))	→	b(c(b(a(x1))))	(1)
b(x1)	→	d(a(x1))	(2)
b(a(c(d(x1))))	→	a(a(a(x1)))	(3)
c(x1)	→	x1	(4)
b(x1)	→	c(d(x1))	(5)

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

c(a(a(x1)))	→	a(b(c(b(x1))))	(6)
b(x1)	→	a(d(x1))	(7)
d(c(a(b(x1))))	→	a(a(a(x1)))	(8)
c(x1)	→	x1	(4)
b(x1)	→	d(c(x1))	(9)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

c^#(a(a(x1)))	→	b^#(c(b(x1)))	(10)
c^#(a(a(x1)))	→	c^#(b(x1))	(11)
c^#(a(a(x1)))	→	b^#(x1)	(12)
b^#(x1)	→	d^#(x1)	(13)
b^#(x1)	→	d^#(c(x1))	(14)
b^#(x1)	→	c^#(x1)	(15)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(x1)	→	c^#(x1)	(15)
c^#(a(a(x1)))	→	b^#(c(b(x1)))	(10)
c^#(a(a(x1)))	→	c^#(b(x1))	(11)
c^#(a(a(x1)))	→	b^#(x1)	(12)

1.1.1.1 Reduction Pair Processor

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

[b^#(x₁)]

-∞

-1	-1	-1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c^#(x₁)]

-∞

-1	-1	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

-1	-1	0
1	0	-1
-1	-1	-∞

· x₁

[c(x₁)]

-∞

0	1	-1
1	0	-1
0	-1	0

· x₁

[b(x₁)]

-1	-1	-1
0	-1	0
1	0	1

· x₁

[d(x₁)]

-∞

-1

-∞	-∞	-1
-∞	-1	-1
-1	-∞	-∞

· x₁

the pair

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

→

c^#(b(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 integers

[b^#(x₁)]

-∞

-∞	1	-1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c^#(x₁)]

-∞

-∞	1	-1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

-∞	-∞	-∞
-∞	0	0
0	-∞	-∞

· x₁

[c(x₁)]

-∞

-1

0	-∞	0
-∞	0	-∞
-∞	2	0

· x₁

[b(x₁)]

-∞

1	1	1
-1	-∞	-1
-1	1	-1

· x₁

[d(x₁)]

-∞

-1	-∞	-1
-1	-∞	-∞
-∞	-∞	-1

· x₁

the pair

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

→

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

(10)

could be deleted.

1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

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

1.1.1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

c^#(a(a(x1)))	→	b^#(x1)	(12)

1	>	1
b^#(x1)	→	c^#(x1)	(15)

1	≥	1

As there is no critical graph in the transitive closure, there are no infinite chains.