Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x05)

The rewrite relation of the following TRS is considered.

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

d(c(x1))	→	c(b(x1))	(7)
b(c(c(x1)))	→	a(c(d(x1)))	(5)
a(x1)	→	c(c(d(x1)))	(6)

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1 Switch to Innermost Termination

The TRS does not have overlaps with the pairs and is locally confluent:

Hence, it suffices to show innermost termination in the following.

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)]

-∞

-∞	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[d^#(x₁)]

-∞

-∞	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

0	0	1
0	0	-∞
-∞	0	0

· x₁

[b^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[d(x₁)]

0	0	1
-∞	-∞	0
-∞	-∞	0

· x₁

[b(x₁)]

-∞	0	0
-∞	0	0
-∞	0	0

· x₁

[a(x₁)]

0	0	1
0	0	1
0	0	1

· x₁

the pair

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

→

d^#(x1)

(11)

could be deleted.

1.1.1.1.1.1.1 Reduction Pair Processor

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

[a^#(x₁)]

-∞

0	0	1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[d^#(x₁)]

-∞

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞

0	0	0
0	0	1
0	-∞	0

· x₁

[b^#(x₁)]

-∞

0	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[d(x₁)]

-∞	-∞	0
-∞	0	1
-∞	-∞	0

· x₁

[b(x₁)]

-∞

0	-∞	-∞
0	-∞	0
0	-∞	0

· x₁

[a(x₁)]

0	0	1
-∞	0	1
-∞	0	1

· x₁

the pair

a^#(x1)

→

d^#(x1)

(12)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.