Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/secr10)

The rewrite relation of the following TRS is considered.

a(a(x1))	→	c(b(a(b(a(x1)))))	(1)
b(a(b(x1)))	→	b(x1)	(2)
a(a(a(x1)))	→	c(c(a(x1)))	(3)
c(c(x1))	→	a(b(c(b(a(x1)))))	(4)
a(c(a(x1)))	→	c(c(a(x1)))	(5)
c(a(c(x1)))	→	a(a(c(x1)))	(6)

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))	→	c^#(b(a(b(a(x1)))))	(7)
a^#(a(x1))	→	b^#(a(b(a(x1))))	(8)
a^#(a(x1))	→	a^#(b(a(x1)))	(9)
a^#(a(x1))	→	b^#(a(x1))	(10)
a^#(a(a(x1)))	→	c^#(c(a(x1)))	(11)
a^#(a(a(x1)))	→	c^#(a(x1))	(12)
c^#(c(x1))	→	a^#(b(c(b(a(x1)))))	(13)
c^#(c(x1))	→	b^#(c(b(a(x1))))	(14)
c^#(c(x1))	→	c^#(b(a(x1)))	(15)
c^#(c(x1))	→	b^#(a(x1))	(16)
c^#(c(x1))	→	a^#(x1)	(17)
a^#(c(a(x1)))	→	c^#(c(a(x1)))	(18)
c^#(a(c(x1)))	→	a^#(a(c(x1)))	(19)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(c(x1))	→	a^#(b(c(b(a(x1)))))	(13)
a^#(a(x1))	→	c^#(b(a(b(a(x1)))))	(7)
c^#(c(x1))	→	c^#(b(a(x1)))	(15)
c^#(c(x1))	→	a^#(x1)	(17)
a^#(a(x1))	→	a^#(b(a(x1)))	(9)
a^#(a(a(x1)))	→	c^#(c(a(x1)))	(11)
c^#(a(c(x1)))	→	a^#(a(c(x1)))	(19)
a^#(a(a(x1)))	→	c^#(a(x1))	(12)
a^#(c(a(x1)))	→	c^#(c(a(x1)))	(18)

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pairs

c^#(c(x1))	→	a^#(b(c(b(a(x1)))))	(13)
a^#(a(x1))	→	c^#(b(a(b(a(x1)))))	(7)
c^#(c(x1))	→	c^#(b(a(x1)))	(15)
c^#(c(x1))	→	a^#(x1)	(17)
a^#(a(x1))	→	a^#(b(a(x1)))	(9)
a^#(a(a(x1)))	→	c^#(a(x1))	(12)

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)]

-∞

-∞	1	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

0	1	0
-∞	0	0
0	1	0

· x₁

[c^#(x₁)]

-∞

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞

0	0	0
0	0	0
0	0	-∞

· x₁

[b(x₁)]

-∞

-∞	0	-∞
-∞	-∞	-∞
-∞	0	-∞

· x₁

the pairs

a^#(a(a(x1)))	→	c^#(c(a(x1)))	(11)
a^#(c(a(x1)))	→	c^#(c(a(x1)))	(18)

could be deleted.

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(c(x1)))	→	a^#(a(c(x1)))	(19)

1	≥	1

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