Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x01)

The rewrite relation of the following TRS is considered.

a(x1)	→	b(b(x1))	(1)
c(b(x1))	→	d(x1)	(2)
e(b(x1))	→	c(c(x1))	(3)
d(b(x1))	→	b(f(x1))	(4)
f(x1)	→	a(e(x1))	(5)
c(x1)	→	x1	(6)
a(a(x1))	→	f(x1)	(7)

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.

c^#(b(x1))	→	d^#(x1)	(8)
e^#(b(x1))	→	c^#(c(x1))	(9)
e^#(b(x1))	→	c^#(x1)	(10)
d^#(b(x1))	→	f^#(x1)	(11)
f^#(x1)	→	a^#(e(x1))	(12)
f^#(x1)	→	e^#(x1)	(13)
a^#(a(x1))	→	f^#(x1)	(14)

1.1 Reduction Pair Processor

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

[c^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

0	0	0
0	0	1
0	-∞	0

· x₁

[d^#(x₁)]

-∞

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[e^#(x₁)]

-∞

0	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞

0	0	0
0	0	0
0	0	0

· x₁

[f^#(x₁)]

-∞

0	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a^#(x₁)]

-∞

0	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[e(x₁)]

-∞

0	-∞	-∞
0	-∞	-∞
0	-∞	0

· x₁

[a(x₁)]

0	0	1
1	0	1
0	0	0

· x₁

[d(x₁)]

-∞

0	-∞	1
0	-∞	1
-∞	-∞	1

· x₁

[f(x₁)]

1	-∞	1
1	-∞	1
0	-∞	0

· x₁

the pair

c^#(b(x1))

→

d^#(x1)

(8)

could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(x1)	→	a^#(e(x1))	(12)
a^#(a(x1))	→	f^#(x1)	(14)

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

[f^#(x₁)]

-∞

0	0	1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a^#(x₁)]

-∞

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[e(x₁)]

-∞

-∞	-∞	0
-∞	0	0
-∞	0	0

· x₁

[a(x₁)]

0	0	1
0	0	0
0	1	0

· x₁

[b(x₁)]

-∞	1	0
-∞	-∞	0
0	0	0

· x₁

[c(x₁)]

0	0	0
0	0	0
0	-∞	0

· x₁

[d(x₁)]

-∞

0	1	0
0	1	0
0	1	0

· x₁

[f(x₁)]

0	1	1
-∞	0	0
0	1	1

· x₁

the pair

f^#(x1)

→

a^#(e(x1))

(12)

could be deleted.

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[a(x₁)]	=	1 · x₁
[f^#(x₁)]	=	1 · x₁
[a^#(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 Size-Change Termination

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

a^#(a(x1))	→	f^#(x1)	(14)

1	>	1

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